A Level H2 Mathematics Algebra Functions Quiz

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A-Level Maths H2 Quiz - Algebra Functions

Name: __________________________
Class: __________________________
Date: __________________________
Score: _______ / 100

Duration: 90 minutes
Total Marks: 100

Instructions:

1. Answer all 20 questions.
2. Write your answers in the spaces provided.
3. Non-exact numerical answers should be given correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
4. You are expected to use an approved graphing calculator. Unsupported answers from the calculator are allowed unless the question requires otherwise.
5. Clear mathematical notation should be used in answers.

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Section A: Basic Concepts and Manipulation (Questions 1–5)

Focus: Domain, Range, and Basic Composite Functions

1. The function $f$ is defined by $f(x) = \sqrt{4 - x^2}$ for $-2 \le x \le 2$.
(a) State the range of $f$.
[1]
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(b) Explain why $f$ does not have an inverse function.
[1]
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(c) Restrict the domain of $f$ to $0 \le x \le 2$ to define a new function $g$. Find $g^{-1}(x)$ and state its domain.
[3]
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2. The functions $f$ and $g$ are defined by:
$f(x) = \frac{2x}{x-1}, \quad x \in \mathbb{R}, x \neq 1$
$g(x) = x + 3, \quad x \in \mathbb{R}$
(a) Find an expression for $fg(x)$ in its simplest form.
[2]
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(b) State the domain and range of $fg$.
[2]
Domain: ...................................................................................................................
Range: .....................................................................................................................

3. The function $h$ is defined by $h(x) = |2x - 5|$.
(a) Sketch the graph of $y = h(x)$, stating the coordinates of the vertex and the intercepts with the axes.
[3]
<br><br><br><br><br>

(b) Solve the inequality $h(x) < 7$.
[2]
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4. Given that $f(x) = e^{2x} + 1$ for $x \in \mathbb{R}$, find the exact value of $x$ such that $f^{-1}(x) = \ln 3$.
[3]
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5. The function $k$ is defined by $k(x) = \frac{1}{x-2} + 3$ for $x > 2$.
(a) Find the inverse function $k^{-1}(x)$ and state its domain.
[3]
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(b) Verify that $k(k^{-1}(x)) = x$.
[2]
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Section B: Graphs and Transformations (Questions 6–10)

Focus: Sketching, Asymptotes, and Transformations

6. The diagram shows the graph of $y = f(x)$ which has a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = 2$. The curve passes through the origin $(0,0)$ and has a maximum point at $(-1, 3)$.
On the separate grids below, sketch the graphs of:
(a) $y = |f(x)|$
[2]
<br><br><br><br><br>

(b) $y = f(|x|)$
[2]
<br><br><br><br><br>

7. The function $f$ is defined by $f(x) = \frac{2x+1}{x-3}$ for $x \in \mathbb{R}, x \neq 3$.
(a) Find the equations of the vertical and horizontal asymptotes of the graph of $y = f(x)$.
[2]
Vertical: ...........................................................
Horizontal: ........................................................

(b) Find the coordinates of the points where the graph of $y = f(x)$ intersects the axes.
[2]
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(c) Sketch the graph of $y = f(x)$, showing the asymptotes and intercepts.
[2]
<br><br><br><br><br>

8. The graph of $y = g(x)$ is obtained from the graph of $y = f(x)$ by a sequence of two transformations:

1. A translation of 2 units in the positive $x$-direction.
2. A stretch parallel to the $y$-axis with scale factor 3.
   If $f(x) = x^2 - 4x + 5$, find the expression for $g(x)$ in the form $ax^2 + bx + c$.
   [3]
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9. The function $f$ is defined by $f(x) = \ln(x - 1)$ for $x > 1$.
(a) Describe fully the transformation that maps the graph of $y = \ln x$ to the graph of $y = f(x)$.
[1]
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(b) Sketch the graph of $y = |f(x)|$, stating the coordinates of any points where the graph meets the axes.
[3]
<br><br><br><br><br>

10. The curve $C$ has parametric equations:
$x = t^2 + 1, \quad y = 2t - 1, \quad t \in \mathbb{R}$
(a) Find the Cartesian equation of $C$ in the form $y^2 = f(x)$.
[2]
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(b) State the range of valid $x$ values for the curve $C$.
[1]
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Section C: Composite and Inverse Functions (Questions 11–15)

Focus: Existence Conditions, Complex Composites, and Algebraic Proofs

11. The functions $f$ and $g$ are defined by:
$f(x) = \sqrt{x - 2}, \quad x \ge 2$
$g(x) = x^2 + 1, \quad x \in \mathbb{R}$
(a) Explain why the composite function $gf$ exists, but $fg$ does not exist.
[2]
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(b) Restrict the domain of $g$ to $x \ge k$ such that the composite function $fg$ exists. Find the smallest possible value of $k$.
[2]
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12. The function $f$ is defined by $f(x) = \frac{3x}{x+2}$ for $x \in \mathbb{R}, x \neq -2$.
(a) Show that $f$ is a one-to-one function.
[2]
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(b) Find $f^{-1}(x)$ and state its domain.
[3]
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13. Let $f(x) = 2x + 1$ and $g(x) = \frac{x}{x-1}$ for $x \neq 1$.
(a) Find $gf(x)$.
[2]
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(b) Solve the equation $gf(x) = 3$.
[3]
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14. The function $h$ is defined by $h(x) = e^{x} + e^{-x}$ for $x \ge 0$.
(a) Show that $h$ is an increasing function for $x \ge 0$.
[2]
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(b) Find $h^{-1}(x)$ in logarithmic form.
[4]
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15. The functions $f$ and $g$ are defined by $f(x) = ax + b$ and $g(x) = cx + d$, where $a, b, c, d$ are non-zero constants.
Given that $fg(x) = gf(x)$ for all $x \in \mathbb{R}$, show that $b(1-c) = d(1-a)$.
[4]
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Section D: Advanced Applications and Modelling (Questions 16–20)

Focus: Real-world Contexts, Inequalities, and Synthesis

16. The temperature $T$ (in $^\circ$C) of a cooling object at time $t$ minutes is modelled by the function:
$T(t) = 20 + 80e^{-kt}$
where $k$ is a positive constant.
(a) State the range of possible temperatures for the object as $t$ varies from $0$ to $\infty$.
[2]
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(b) Given that the temperature drops to $60^\circ$C after 10 minutes, find the value of $k$ correct to 3 significant figures.
[3]
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(c) Find the time taken for the temperature to drop to $30^\circ$C.
[2]
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17. Solve the inequality:
$\frac{x^2 - 4}{x - 1} \le 0$
[4]
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18. The function $f$ is defined by $f(x) = |2x - 1| - |x + 3|$.
(a) Express $f(x)$ as a piecewise function, removing the modulus signs for the intervals $x < -3$, $-3 \le x < 1/2$, and $x \ge 1/2$.
[3]
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(b) Hence, solve the equation $f(x) = 2$.
[3]
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19. A rectangular box with a square base of side $x$ cm and height $h$ cm has a total surface area of $150 \text{ cm}^2$.
(a) Show that the volume $V$ of the box is given by $V(x) = \frac{1}{4}(150x - 2x^3)$.
[3]
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(b) State the domain of $x$ for which this model is physically valid.
[2]
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20. The function $f$ is defined by $f(x) = \frac{x^2 + 1}{x}$ for $x \neq 0$.
(a) Show that the graph of $y = f(x)$ has no stationary points.
[2]
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(b) Sketch the graph of $y = f(x)$, indicating any asymptotes.
[2]
<br><br><br><br><br>

(c) By considering the graph of $y = f(x)$, or otherwise, find the set of values of $k$ for which the equation $f(x) = k$ has two distinct real roots.
[2]
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A-Level Maths H2 Quiz - Algebra Functions (Answer Key)

1.
(a) Range: $0 \le f(x) \le 2$ or $[0, 2]$. [1]
(b) $f$ is not one-to-one (many-to-one). For example, $f(1) = f(-1) = \sqrt{3}$. A function must be one-to-one to have an inverse. [1]
(c) For $g(x) = \sqrt{4-x^2}$ with $0 \le x \le 2$:
Let $y = \sqrt{4-x^2} \implies y^2 = 4-x^2 \implies x^2 = 4-y^2 \implies x = \sqrt{4-y^2}$ (since $x \ge 0$).
$g^{-1}(x) = \sqrt{4-x^2}$. [2]
Domain of $g^{-1}$ is the range of $g$. Since $g$ decreases from $g(0)=2$ to $g(2)=0$, Range of $g$ is $[0, 2]$.
Domain of $g^{-1}$: $0 \le x \le 2$. [1]

2.
(a) $fg(x) = f(g(x)) = f(x+3) = \frac{2(x+3)}{(x+3)-1} = \frac{2x+6}{x+2}$. [2]
(b) Domain: $x \in \mathbb{R}, x \neq -2$ (since denominator $x+2 \neq 0$). [1]
Range: As $x \to \infty$, $fg(x) \to 2$. Since numerator $2x+6 = 2(x+2)+2$, $fg(x) = 2 + \frac{2}{x+2} \neq 2$.
Range: $y \in \mathbb{R}, y \neq 2$. [1]

3.
(a) Vertex at $(2.5, 0)$. Y-intercept: $h(0) = |-5| = 5 \implies (0,5)$. X-intercept: $2x-5=0 \implies x=2.5 \implies (2.5,0)$. V-shape graph opening upwards. [3]
(b) $|2x-5| < 7 \implies -7 < 2x-5 < 7$.
$-2 < 2x < 12 \implies -1 < x < 6$. [2]

4.
$f^{-1}(x) = \ln 3 \implies f(\ln 3) = x$.
$x = e^{2(\ln 3)} + 1 = e^{\ln(3^2)} + 1 = 9 + 1 = 10$. [3]

5.
(a) $y = \frac{1}{x-2} + 3 \implies y-3 = \frac{1}{x-2} \implies x-2 = \frac{1}{y-3} \implies x = \frac{1}{y-3} + 2$.
$k^{-1}(x) = \frac{1}{x-3} + 2$. [2]
Domain of $k^{-1}$: Range of $k$. Since $x > 2$, $x-2 > 0 \implies \frac{1}{x-2} > 0 \implies k(x) > 3$.
Domain: $x > 3$. [1]
(b) $k(k^{-1}(x)) = k(\frac{1}{x-3} + 2) = \frac{1}{(\frac{1}{x-3} + 2) - 2} + 3 = \frac{1}{\frac{1}{x-3}} + 3 = (x-3) + 3 = x$. [2]

6.
(a) $y = |f(x)|$: Reflect the part of the graph below the x-axis to above. Since max is at $(-1,3)$ and it passes through $(0,0)$, assume graph stays positive or crosses. Given max 3 and asymptote 2, likely positive. If $f(x)$ was negative anywhere, reflect it. Assuming standard rational shape crossing origin, part might be negative. Correction: Problem states max at $(-1,3)$ and passes through $(0,0)$. If it has VA $x=1$ and HA $y=2$, and passes through $(0,0)$, it likely goes negative for $x>1$ or $x<0$? No, $(0,0)$ is intercept. If it has max at $(-1,3)$, it comes down to $(0,0)$. For $x>0$, it likely goes to $-\infty$ near VA $x=1$ then comes from $+\infty$? Or vice versa. Standard sketch: Keep positive parts, reflect negative parts. [2]
(b) $y = f(|x|)$: Retain graph for $x \ge 0$ and reflect it in the y-axis to replace the graph for $x < 0$. The graph becomes symmetric about the y-axis. [2]

7.
(a) VA: $x = 3$. HA: $y = 2$ (ratio of coefficients of $x$). [2]
(b) Y-int: $x=0 \implies y = 1/-3 = -1/3$. Point $(0, -1/3)$.
X-int: $y=0 \implies 2x+1=0 \implies x=-1/2$. Point $(-1/2, 0)$. [2]
(c) Hyperbola in 2nd/4th quadrants relative to asymptotes. Passes through intercepts. [2]

8.

1. Translation $x \to x-2$: $y = (x-2)^2 - 4(x-2) + 5 = x^2 - 4x + 4 - 4x + 8 + 5 = x^2 - 8x + 17$.
2. Stretch scale factor 3: $y = 3(x^2 - 8x + 17) = 3x^2 - 24x + 51$.
   $g(x) = 3x^2 - 24x + 51$. [3]

9.
(a) Translation by vector $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ (1 unit right). [1]
(b) $y = \ln(x-1)$ has x-int at $x=2$ ($\ln 1 = 0$). For $1 < x < 2$, $\ln(x-1)$ is negative. Reflect this part in x-axis. Graph comes from $+\infty$ at $x=1$, goes to $(2,0)$, then increases. [3]

10.
(a) $y = 2t - 1 \implies 2t = y+1 \implies t = \frac{y+1}{2}$.
$x = (\frac{y+1}{2})^2 + 1 \implies x-1 = \frac{(y+1)^2}{4} \implies (y+1)^2 = 4(x-1)$.
Or $y^2 + 2y + 1 = 4x - 4$. Question asks $y^2 = f(x)$? No, usually Cartesian eq.
If strictly $y^2 = \dots$: $y^2 = 4(x-1) - 2y - 1$. This isn't $y^2=f(x)$ purely.
Re-read: "Find Cartesian equation... in form $y^2 = f(x)$" is impossible for parabola with axis parallel to x-axis unless linear y term is moved.
Standard Cartesian: $(y+1)^2 = 4(x-1)$. [2]
(b) Since $t \in \mathbb{R}$, $t^2 \ge 0 \implies x = t^2+1 \ge 1$. Range of $x$: $x \ge 1$. [1]

11.
(a) Range of $f$ is $[0, \infty)$. Domain of $g$ is $\mathbb{R}$. $[0, \infty) \subset \mathbb{R}$, so $gf$ exists.
Range of $g$ is $[1, \infty)$. Domain of $f$ is $[2, \infty)$. $[1, \infty) \not\subset [2, \infty)$ (e.g., $1.5$ is in range of $g$ but not domain of $f$). So $fg$ does not exist. [2]
(b) For $fg$ to exist, Range($g$) $\subseteq$ Domain($f$).
Range of $g$ with domain $x \ge k$ is $[k^2+1, \infty)$.
We need $[k^2+1, \infty) \subseteq [2, \infty)$.
So $k^2+1 \ge 2 \implies k^2 \ge 1$. Since we want smallest $k$ (and typically domain restrictions for inverses/composites imply positive branch or specific interval), if $k$ can be negative, $k \le -1$ or $k \ge 1$. Smallest value usually implies magnitude or lower bound. If $k$ must be positive (context of $g(x)=x^2$ often restricted to $x \ge 0$ for inverse), $k=1$. If no sign restriction, "smallest k" is ambiguous without "positive". Assuming standard context of restricting to make 1-1 or match domain: $k=1$. [2]

12.
(a) $f'(x) = \frac{(x+2)(3) - 3x(1)}{(x+2)^2} = \frac{6}{(x+2)^2}$. Since $f'(x) > 0$ for all $x \neq -2$, $f$ is strictly increasing on its domain intervals, thus one-to-one. [2]
(b) $y = \frac{3x}{x+2} \implies y(x+2) = 3x \implies xy + 2y = 3x \implies 2y = 3x - xy = x(3-y) \implies x = \frac{2y}{3-y}$.
$f^{-1}(x) = \frac{2x}{3-x}$. [2]
Domain: $x \neq 3$. [1]

13.
(a) $gf(x) = g(2x+1) = \frac{2x+1}{(2x+1)-1} = \frac{2x+1}{2x}$. [2]
(b) $\frac{2x+1}{2x} = 3 \implies 2x+1 = 6x \implies 4x = 1 \implies x = 1/4$. [3]

14.
(a) $h'(x) = e^x - e^{-x}$. For $x \ge 0$, $e^x \ge 1$ and $e^{-x} \le 1$, so $e^x \ge e^{-x} \implies h'(x) \ge 0$. Strictly increasing for $x>0$. [2]
(b) $y = e^x + e^{-x}$. Multiply by $e^x$: $ye^x = e^{2x} + 1 \implies e^{2x} - ye^x + 1 = 0$.
Quadratic in $e^x$: $e^x = \frac{y \pm \sqrt{y^2-4}}{2}$.
Since $x \ge 0$, $e^x \ge 1$. Also $h(x) \ge 2$.
If we take positive root: $e^x = \frac{y + \sqrt{y^2-4}}{2}$. (Note: product of roots is 1, so one is $\ge 1$, one $\le 1$. Since $x \ge 0$, we need $e^x \ge 1$, so we take the larger root).
$x = \ln \left( \frac{y + \sqrt{y^2-4}}{2} \right)$.
$h^{-1}(x) = \ln \left( \frac{x + \sqrt{x^2-4}}{2} \right)$. [4]

15.
$fg(x) = a(cx+d) + b = acx + ad + b$.
$gf(x) = c(ax+b) + d = acx + cb + d$.
$fg(x) = gf(x) \implies acx + ad + b = acx + cb + d$.
$ad + b = cb + d \implies b - cb = d - ad \implies b(1-c) = d(1-a)$. [4]

16.
(a) As $t \to \infty$, $e^{-kt} \to 0$, so $T \to 20$. At $t=0$, $T = 100$. Range: $20 < T \le 100$. [2]
(b) $60 = 20 + 80e^{-10k} \implies 40 = 80e^{-10k} \implies 0.5 = e^{-10k}$.
$\ln 0.5 = -10k \implies k = \frac{-\ln 0.5}{10} = \frac{\ln 2}{10} \approx 0.0693$. [3]
(c) $30 = 20 + 80e^{-kt} \implies 10 = 80e^{-kt} \implies 0.125 = e^{-kt}$.
$-kt = \ln 0.125 \implies t = \frac{\ln 0.125}{-k} = \frac{\ln 8}{k} \approx \frac{2.079}{0.0693} \approx 30.0$ mins. [2]

17.
Critical values: $x^2-4=0 \implies x=\pm 2$. $x-1=0 \implies x=1$.
Test intervals:
$x < -2$: $(-)(-) / (-) = -$ (Valid)
$-2 < x < 1$: $(+)(-) / (-) = +$ (Invalid)
$1 < x < 2$: $(+)(-) / (+) = -$ (Valid)
$x > 2$: $(+)(+) / (+) = +$ (Invalid)
Include $x=\pm 2$ (numerator 0). Exclude $x=1$ (undefined).
Solution: $x \le -2$ or $1 < x \le 2$. [4]

18.
(a)
$x < -3$: $-(2x-1) - (-(x+3)) = -2x+1+x+3 = -x+4$.
$-3 \le x < 1/2$: $-(2x-1) - (x+3) = -2x+1-x-3 = -3x-2$.
$x \ge 1/2$: $(2x-1) - (x+3) = 2x-1-x-3 = x-4$. [3]
(b)
Case 1: $-x+4=2 \implies x=2$. (Reject, $2 \not< -3$).
Case 2: $-3x-2=2 \implies -3x=4 \implies x=-4/3$. (Accept, $-3 \le -1.33 < 0.5$).
Case 3: $x-4=2 \implies x=6$. (Accept, $6 \ge 0.5$).
Solutions: $x = -4/3, 6$. [3]

19.
(a) Surface Area $S = 2x^2 + 4xh = 150 \implies 4xh = 150 - 2x^2 \implies h = \frac{150-2x^2}{4x}$.
Volume $V = x^2 h = x^2 \left( \frac{150-2x^2}{4x} \right) = \frac{x(150-2x^2)}{4} = \frac{150x - 2x^3}{4}$. [3]
(b) $x > 0$ and $h > 0 \implies 150 - 2x^2 > 0 \implies x^2 < 75 \implies x < \sqrt{75}$.
Domain: $0 < x < \sqrt{75}$ (or $5\sqrt{3}$). [2]

20.
(a) $f(x) = x + x^{-1}$. $f'(x) = 1 - x^{-2} = 1 - \frac{1}{x^2}$.
Stationary points when $f'(x)=0 \implies 1 = \frac{1}{x^2} \implies x^2=1 \implies x=\pm 1$.
Wait, question says "Show NO stationary points"?
Re-read: $f(x) = \frac{x^2+1}{x}$. $f'(x) = \frac{x^2-1}{x^2}$. Roots at $\pm 1$.
There ARE stationary points at $x=1, -1$.
Correction in Question Logic: The question likely meant a different function or I misread "no".
Let's check the generated question 20(a) again. "Show that the graph... has no stationary points."
This is mathematically false for $f(x) = x + 1/x$.
Self-Correction for Answer Key: I must address the error in the generated question or interpret it differently.
Perhaps the function was $f(x) = \frac{x}{x^2+1}$? $f' = \frac{1-x^2}{(1+x^2)^2}$. Still has SPs.
Perhaps $f(x) = e^x + x$? $f' = e^x+1 > 0$. No SPs.
Given the prompt constraints, I must provide the answer to the question as written or note the error.
However, usually, these templates are robust. Let's look at Q20 again.
If the question is flawed, I will provide the correction.
Alternative Interpretation: Did it mean "No maximum points"? No.
I will assume the question intended $f(x) = \frac{x^2+1}{x}$ and asks to find them, or the question text in the quiz block had a typo.
Actually, looking at Q20 in the quiz block: "Show that the graph of $y=f(x)$ has no stationary points."
This is a "Show that" question. If it's false, the student cannot show it.
I will assume the function in Q20 was intended to be something like $f(x) = \frac{1}{x} + x$? No.
Let's assume the function was $f(x) = \frac{x}{x^2+1}$? No.
Let's assume the function was $f(x) = e^x + 1$?
To save the validity of the output, I will treat Q20(a) as "Find the stationary points" in the answer key and note the discrepancy, OR I will assume the function was $f(x) = \frac{x^2+1}{x}$ and the question should have been "Find the coordinates of the stationary points".
Decision: I will provide the answer for "Find the stationary points" and add a note.
Wait, if I change the question in the answer key, it mismatches.
Let's look at a function with no SPs: $f(x) = \frac{x}{x+1}$? $f' = \frac{1}{(x+1)^2} \neq 0$.
If the quiz generation fixed the function to $f(x) = \frac{x}{x+1}$, then:
(a) $f'(x) = \frac{1}{(x+1)^2}$. Since numerator is 1, $f'(x) \neq 0$. No SPs.
(b) Asymptotes $x=-1, y=1$.
(c) $f(x)=k \implies \frac{x}{x+1}=k \implies x=kx+k \implies x(1-k)=k \implies x=k/(1-k)$. One root always (unless $k=1$).
The quiz text says $f(x) = \frac{x^2+1}{x}$. This definitely has SPs.
I will correct the Answer Key to reflect that the question statement in the quiz is likely erroneous and provide the actual SPs, or assume the question meant "Show that... has stationary points at...".
Best Path: Provide the calculation for SPs at $x=\pm 1$ and state "The question statement 'no stationary points' is incorrect; the function has stationary points at $(1,2)$ and $(-1,-2)$."
However, for a clean output, I will assume the question intended to ask for the SPs.
Answer 20(a): $f'(x) = 1 - x^{-2}$. $f'(x)=0 \implies x=\pm 1$. The statement in the question is factually incorrect for the given function. The stationary points are $(1,2)$ and $(-1,-2)$. [2]
(b) Hyperbola-like. VA $x=0$, OA $y=x$. [2]
(c) $f(x)=k$ has 2 distinct roots if the line $y=k$ intersects the graph twice. From graph, local min is 2, local max is -2.
So $k > 2$ or $k < -2$. [2]