Functions — Free Maths AI HL Practice Questions

Find $f^{-1}(x)$, the inverse function of $f$. [3 marks]

The composite function $f \circ f$ is defined on a suitable restricted domain. Show that $f(f(x)) = x$, and hence explain what this result reveals about the relationship between $f$ and $f^{-1}$. Total: 8 marks (Note: total adjusted to 8 to match independently markable steps; original 5-mark allocation was insufficient for the cognitive arc.) [3 marks]

Hence find an expression for $t$ in terms of $V$, stating its domain. [3 marks]

The volume of the tank is $V = \pi \displaystyle\int_0^4 [f(x)]^2\, dx$. Find $V$, giving your answer correct to 3 significant figures. [3 marks]

The company modifies the design using $g(x) = \dfrac{12}{x+2} - 2 + k$, where $k$ is a positive constant, so that the new tank holds exactly twice the volume of the original tank. (i) Show that $k$ satisfies the equation $4k^2 + (24\ln 3 - 16)k + (48\ln 3 - 64) = 0$. [2]
(ii) Find the value of $k$, justifying why only one root of the equation is valid. [1 mark]

Show that the range of $g$ is $[1, \infty)$. [2 marks]

The domain of $g$ is restricted to $x \geq 2$. Find an expression for $g^{-1}(x)$, stating its domain. [2 marks]

Determine the coordinates of the vertex of $g(x)$. [2 marks]

The graph of $g(x)$ is translated by $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$. Write down the equation of the resulting function $h(x)$ in the form $y = a(x - h)^2 + k$, and hence state the range of $h(x)$. [2 marks]

Write down the coordinates of the vertex of the graph of $h$. [1 mark]

The graph of $h$ is translated by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ so that the resulting graph has exactly one $x$-intercept. Determine the value of $k$. [1 mark]

The function $g$ is defined by $g(x) = -f(x+2) + 3$. Determine the coordinates of the minimum point of $g$. [2 marks]

The graph of $f$ is transformed as follows: - reflected in the $y$-axis, - translated by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$, - stretched vertically by scale factor $4$. Determine the equation of the resulting function $h(x)$, writing your answer in the form $h(x) = a\ln(x^2 + bx + c) + d$, where $a,\, b,\, c,\, d \in \mathbb{R}$. [4 marks]

The graph of $f$ is reflected in the $x$-axis and then stretched vertically by a scale factor of $2$. Write down the equation of the resulting function $g(x)$. [2 marks]

Find all values of $x$ in the interval $0 \leq x \leq 2\pi$ for which $g(x) = 1$. [2 marks]

State the value of $a$. [1 mark]

Given that the Ferris wheel completes one full revolution in $8$ minutes, find the value of $b$. [3 marks]

Find the other time between midnight and 12:00, apart from 04:00, when the depth of water equals the value found in part (a). [3 marks]

The turbine requires a minimum water height of $10$ metres to operate. Determine the total number of hours in the first $24$-hour period during which the turbine is operational. [4 marks]

Calculate the depth of water at 2:00 am. [2 marks]

A boat requires a minimum depth of $2.5\,\text{m}$ to enter the harbour safely. Determine the first time after midnight at which the depth is exactly $2.5\,\text{m}$, and hence find the latest time before this that the boat could have entered the harbour. [2 marks]

Calculate the value of $r$ that minimises the total cost. Give your answer correct to 3 significant figures. [2 marks]

Hence determine the minimum total cost and justify that this value of $r$ gives a minimum rather than a maximum. [1 mark]

Find the inverse function $f^{-1}(x)$, stating its domain. Hence state the range of $f$. [3 marks]

Show that $h(0) = 12$, and hence determine the range of $h(t)$ for $t \geq 0$. [3 marks]

The drone's flight is considered stable when its height satisfies $3.5 \leq h \leq 4.5$. Determine the time interval, correct to 3 significant figures, during which the drone is flying at a stable height. [2 marks]

Hence state the range of $C(t)$ for $t \geq 0$. [1 mark]

The medication is considered effective when the concentration is at least $1.5\,\text{mg/L}$. Determine, correct to 3 significant figures, the total duration for which the medication is effective. [3 marks]

The graph of $g$ is then reflected in the $x$-axis to give the graph of a function $p$. Write down $p(x)$ in the form $-(x - h)^2 + k$. [1 mark]

Find the $x$-coordinates of the points where the graph of $p$ intersects the graph of $f$. [2 marks]

The point $B(3,\,-2)$ maps to a point $B'$ on the graph of $g$. (i) Show that $B'$ has coordinates $\left(-\dfrac{1}{2},\,-1\right)$. [2 marks]
(ii) Hence determine whether $B'$ is a local maximum or local minimum of $g$, justifying your answer. [1 mark]

State the domain and range of $g$. [2 marks]

The graph of $g$ intersects the graph of $h(x) = \sqrt{x + k}$, where $k > 0$, at exactly one point. Determine the value of $k$ and the coordinates of the point of intersection. [2 marks]

The point $Q$ on the graph of $f$ has coordinates $\left(a,\, \dfrac{1}{a}\right)$. Write down the coordinates of the image of $Q$ on the graph of $g$, in terms of $a$. [2 marks]

Find all values of $a$ such that the image point found in part (b) lies on the line $y = x$. Give your answers correct to $3$ significant figures. [2 marks]

Calculate the first time $t > 0$ when the height of $P$ is exactly $11$ metres above the water level. Give your answer correct to 3 significant figures. [3 marks]

A second point $Q$ on the rim is always diametrically opposite to $P$. Its height is modelled by $g(t) = 6\sin\!\left(\dfrac{\pi}{30}t + \pi\right) + 10$. Using the identity $\sin(A + B) = \sin A \cos B + \cos A \sin B$, show that $g(t) = -6\sin\!\left(\dfrac{\pi}{30}t\right) + 10$, and hence find all times in the interval $0 \leq t \leq 60$ when $P$ and $Q$ are at the same height. [4 marks]

Hence find the maximum depth of water in the harbour and determine the first time after midnight at which this maximum depth occurs. Give your answer correct to the nearest minute. The following identity is provided: $a\sin\theta + b\cos\theta = R\sin(\theta + \phi)$, where $R = \sqrt{a^2+b^2}$ and $\tan\phi = \dfrac{b}{a}$. [3 marks]

State the maximum voltage. [1 mark]

Find the smallest positive value of $t$ at which the maximum voltage occurs. Give your answer correct to 4 significant figures. [2 marks]

Calculate the temperature at 16:00. [2 marks]

A gardener knows that plants grow best when the temperature is above $12\,^\circ\text{C}$ but below $20\,^\circ\text{C}$. Calculate the total number of hours in the 24-hour period during which the temperature is in this optimal range. [2 marks]

The function $g$ is defined by restricting the domain of $f$ to $[k, \infty)$ for some constant $k \in \mathbb{R}$, with the codomain simultaneously restricted to the range of $g$, so that $$g : [k, \infty) \longrightarrow \text{range of } g, \qquad g(x) = f(x).$$ Determine the complete set of values of $k$ for which $g$ is a bijection (both injective and surjective). Justify your answer fully. [5 marks]

Prove that $P$ is surjective (onto) when its codomain is restricted to $(20, 100)$, that is, $P: [0, \infty) \to (20, 100)$. [3 marks]

Given the results of parts (a) and (b), find $P^{-1}(80)$, giving your answer correct to 3 significant figures. The following information may be used: - Logistic model: $P(t) = \dfrac{L}{1 + Ae^{-kt}}$ - Injective: $f(a) = f(b) \Rightarrow a = b$ - Surjective: for every $y$ in the codomain, there exists $x$ in the domain such that $f(x) = y$ - $\dfrac{d}{dt}\!\left(e^{at}\right) = ae^{at}$ [3 marks]

The function $h$ is defined by $h(x) = f(2x) + 1$. (i) Show that $h(x) = 4(x-1)^2$. [2]
(ii) Determine a sequence of two transformations applied to $f$ that produces $h$. State each transformation precisely. [2 marks]

The claim is made: *"For any quadratic function $f(x) = a(x-h)^2 + k$, reflecting the graph in the $x$-axis maps the vertex $V(h,\, k)$ to the point $(h,\, -k)$."* (i) Verify this claim for $f(x) = (x-2)^2 - 1$. [1]
(ii) Prove the claim for the general case $f(x) = a(x-h)^2 + k$, where $a \neq 0$. [2 marks]

A point $P = (x_0,\, y_0)$ lies on $f$. Show that, under the sequence of transformations found in part (a), $P$ maps to $$P' = \left(\frac{x_0}{b} + \frac{c}{b},\; ay_0 + d\right).$$ Hence explain why the horizontal translation must be applied before the horizontal stretch for the mapping to be consistent with the formula above. [3 marks]

For the specific case $g(x) = 2\sqrt{3x - 6} + 1$: - (i) Find the image of the point $(4,\, 2)$ on $f$ under $T$. [1 mark] -
(ii) The point $Q = \!\left(\dfrac{7}{3},\, 3\right)$ is claimed to lie on $g$ and to be the image of $(1, 1)$ on $f$ under $T$. Verify both claims algebraically, and explain what this result illustrates about how $T$ acts on the graph of $f$. [3 marks]

A second rider boards the wheel $15$ seconds after the first rider. Let $h_2(t)$ represent the height of the second rider above the ground at time $t$ seconds, where $t = 0$ is the moment the first rider boards. (i) Explain why $h_2(t) = h(t - 15)$. [2]
(ii) Find all times in the interval $0 < t \leq 60$ when both riders are at the same height. [4]
(iii) The first rider claims that, over one complete revolution ($0 \leq t \leq 60$), the two riders spend more time with the second rider higher than the first rider than vice versa. Determine whether this claim is correct, justifying your answer. [3 marks]

Using the values given, determine $R$ and $\alpha$, giving your answers correct to three significant figures. [2 marks]

A student claims that the first maximum angular displacement occurs at the time $t^*$ satisfying $\cos(\omega t^* - \alpha) = 1$. (i) Find the value of $t^*$ predicted by the student's method. [1 mark]
(ii) By differentiating $\theta(t)$, find the time $t_{\max}$ at which the first maximum of $\theta(t)$ actually occurs, giving your answer correct to three significant figures. [3 marks]
(iii) Hence evaluate the validity of the student's claim, comparing $t^*$ and $t_{\max}$ and explaining why the two values differ. [1 mark]