Functions, domain, range: notes and practice questions

A tech startup's daily operational costs are modeled by two functions based on the number of new user sign-ups, $x$. Let $C_1(x)$ be the cost for server maintenance and $C_2(x)$ be the cost for customer support. $x$ represents the deviation from the average number of sign-ups, where positive $x$ means more than average and negative $x$ means fewer than average.

(a) The server maintenance cost function is given by $C_1(x) = 3x + 5$, where $x \in \mathbb{Z}$. Find $C_1(-3)$.

(b) The customer support cost function is given by $C_2(x) = 2x^2 - 5x + 7$, where $x \in \mathbb{Z}$. Find $C_2(-3)$.

Consider the function $g(x) = \frac{ax^2+x+k}{x-4}$.

The graph of $y=g(x)$ passes through the point $(1, -2)$ and has an oblique asymptote with equation $y = -3x-11$.

(a) Write down the equation of the vertical asymptote.

(b) Find the value of:

(i) $a$

(ii) $k$

(c) Hence, find the exact coordinates of any points where the graph of $y=g(x)$ intersects the x-axis.

A rectangle is inscribed in an ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The sides of the rectangle are parallel to the coordinate axes. The vertices of the rectangle are located at $(\pm x, \pm y)$, where $x > 0$ and $y > 0$.

(a) Show that the area of the rectangle, $A$, can be expressed as $A = \frac{12x}{5}\sqrt{25-x^2}$.

(b) Show that $\frac{dA}{dx} = \frac{12(25-2x^2)}{5\sqrt{25-x^2}}$.

(c) Hence, find the exact dimensions of the rectangle with the maximum possible area.

The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = -2x + 9$ is the tangent to the graph of $f$ at $x = 3$.

(a) Write down the value of $f'(3)$.

(b) Find $f(3)$.

The function $g$ is defined for all $x \in \mathbb{R}$ where $g(x) = x^2 - 1$ and $h(x) = f(g(x) )$.

(c) Find $h(2)$.

(d) Hence, find the equation of the tangent to the graph of $h$ at $x = 2$.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

Consider the function $h(x) = \frac{8-2x}{x+1}$.

(a) State the largest possible domain for the function $h$.

(b) State the largest possible range for the function $h$.

(c) Find the coordinates of any points where the curve $y = h(x)$ intersects the $x$ and $y$-axes.

A function, $g$, has its derivative given by $g'(x) = -2x^2 + 8x + k$, where $k \in \mathbb{R}$. The following diagram shows part of the graph of $g'$.

The graph of $g'$ has an axis of symmetry $x = q$.

(a) Find the value of $q$.

(b) The vertex of the graph of $g'$ has a y-coordinate of 10. Find the value of $k$.

(c) Find the equation of the tangent to the graph of $g'$ at $x = 0$.

The graph of $g$ has a point of inflexion at $x = c$.

(d) (i) Find the value of $c$.

(ii) Find the values of $x$ for which the graph of $g$ is concave-up. Justify your answer.

Consider the function $f(x) = a^x$ where $x, a \in \mathbb{R}$ and $a > 1$. The graph of $f$ contains the point $(\frac{3}{2}, 27)$.

(a) Show that $a = 9$.

(b) Write down an expression for $f^{-1}(x)$.

(c) Find the value of $f^{-1}(\frac{1}{81})$.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(i) Show that $8, p, q$ and $27$ are four consecutive terms in a geometric sequence.

Consider the arithmetic sequence $\log_9 8, \log_9 p, \log_9 q, \log_9 27$, where $p > 1$ and $q > 1$.

(ii) Find the value of $p$ and the value of $q$.

For each of the following functions, write down its greatest possible domain and range.

(a) $f(x) = \frac{5}{2x-8}$

(b) $g(x) = \frac{10x-1}{2x-8}$

(c) $h(x) = \frac{10x-1}{2x}$

(d) $p(x) = \frac{5}{2x}$

Solve the inequality $3x^2 + 5x - 2 > 0$.

The function $g$ is defined by $g(x) = \sqrt{3x^2 + 5x - 2}$, where $x \in \mathbb{R}, x \ge k$.

Find the least value of $k$ for which $g^{-1}$ exists, justifying your answer.

A landscape architect is designing a section of a garden path. The shape of one edge of the path can be modelled by the function $h(x) = \frac{1}{4}x^2 + \frac{1}{2}$ for $x \ge 0$, where $x$ and $h(x)$ are measured in metres.

(a) (i) Find $h^{-1}(x)$, the inverse of $h(x)$, and state its domain.

(ii) Write down the range of $h^{-1}(x)$.

(b) The graph of $h$ intersects the graph of $h^{-1}$ at two points. Find the $x$ -coordinates of these two points.

(c) Find the area enclosed by the graph of $h$ and the graph of $h^{-1}$.

(d) Find $h'(x)$.

(e) Find the value of $x$ for which the graph of $h$ and the graph of $h^{-1}$ have the same gradient.

A function is defined by $f(x) = \frac{ax+b}{x+d}$.

(a) The asymptotes of the graph of $y=f(x)$ are a vertical line and a horizontal line which intersect at the point $(-2, 4)$. Find the value of $a$ and the value of $d$.

(b) The graph of $y=f(x)$ passes through the point $(0, -3)$. Find the value of $b$.

Consider the function $g(x) = \csc\left(x+\frac{\pi}{3}\right)$, for $0\le x\le \frac{\pi}{3}$.

Determine the range of $g$.

The region bounded by the graph of $y = g(x)$, the $x$-axis and the lines $x = 0$ and $x = \frac{\pi}{3}$ is rotated $2\pi$ radians about the $x$-axis.

Find the volume of the solid generated.

A team of engineers is testing two autonomous robots, Alpha and Beta, on a straight track. Their positions are measured as the distance from a fixed starting point. The experiment runs for 10 minutes.

The position of Robot Alpha, $P_A$ metres, at time $t$ minutes can be modelled by the function $P_A(t) = 3\sin(2t + 5) + 14t + 20$, where $0 \le t \le 10$.

The position of Robot Beta, $P_B$ metres, at time $t$ minutes can be modelled by the function $P_B(t) = 12t + 25$, where $0 \le t \le 10$.

Use the engineers' models to find the initial position of

(i) Robot Beta;

(ii) Robot Alpha correct to three significant figures.

Find the values of $t$ when Robot Alpha and Robot Beta are at the same position. Give your answers correct to three significant figures.

For $t > 5$, prove that Robot Alpha was always ahead of Robot Beta.

For $0 \le t \le 10$, find the total amount of time when the speed of Robot Beta was greater than the speed of Robot Alpha. Give your answer correct to three significant figures.

Consider the function $g(x) = -2(x + 1)(x - p)$, where $x \in \mathbb{R}$ and $p$ is a real constant.

The axis of symmetry of the graph of $g$ has equation $x = 3$.

Show that $p = 7$.

Find the coordinates of the vertex of the graph of $g$.

Hence, write down the range of $g$.

A tech company's daily profit, $P$, in thousands of dollars, from producing $x$ units of a new gadget is modelled by the function $f(x) = -2x^2 + 16x + 468$, for $x \in \mathbb{R}$.

(a) Find the range of the company's daily profit.

Due to new environmental regulations, the company faces a levy that adjusts its profit. The adjusted profit, $A$, is given by the function $g(P) = P + k$, where $k \in \mathbb{R}$ is a constant representing the levy's impact.

Given that the adjusted profit $(g \circ f)(x)$ must be non-positive for all $x \in \mathbb{R}$, determine the set of possible values for $k$.

A designer is creating a decorative glass container shaped like a dome. The outer profile of the container can be modelled by the function $f(x) = \sqrt{9-x^2}$, where $0 \le x \le 3$ and $x$ and $y$ are measured in metres.

Sketch the curve $y = f(x)$, clearly indicating the coordinates of the endpoints.

Show that the inverse function of $f$ is given by $f^{-1}(x) = \sqrt{9-x^2}$.

State the domain and range of $f^{-1}$.

The container is formed by rotating the curve $y = f(x)$ by $2\pi$ about the y-axis. Show that the volume, $V \text{ m}^3$, of liquid in the container when it is filled to a height of $h$ metres is given by $V = \pi \left( 9h - \frac{1}{3}h^3 \right)$.

Hence, determine the maximum volume of the container.

At $t = 0$, the container is empty. Liquid is then added to the container at a constant rate of $0.5 \text{ m}^3\text{s}^{-1}$.

Find the time it takes to fill the container to its maximum volume.

Find the rate of change of the height of the liquid when the container is filled to half its maximum volume.

The population, $P$, in hundreds of plants, $x$ years after the study began, is modeled by the function $f(x) = -2x^2 + 8x + 5$, for $x \in \mathbb{R}$.

(a) Find the range of $f$.

Due to environmental changes, a new invasive plant species is introduced. The impact of this invasive species on the orchid's population is modeled by a function $g(P) = -P + k$, where $P$ is the current orchid population (in hundreds of plants) and $k$ is a constant representing the environmental resilience. For the orchid species to survive and thrive, the combined effect $(g \circ f)(x)$ must always be non-negative (i.e., $(g \circ f)(x) \ge 0$) for all $x \in \mathbb{R}$. Determine the set of possible values for $k$.

Consider the functions $f(x) = 5 - x^2$, $g(x) = \frac{3}{x-2}$, and $h(x) = e^x + 1$.

(a) Find the range of $f(x)$.

(b) Find the range of $g(x)$.

(c) Find the range of $h(x)$.

(d) Find an expression for $(g \circ f)(x)$.

(e) Solve the equation $(g \circ f)(x) = -1$.

(f) Solve the inequality $(h \circ f)(x) < e+1$.