Composite functions, inverse functions: notes and practice questions

Consider the function $f(x) = \frac{7x + 6}{5x - 3}$, for $x \neq 3$.

$a$ Find the $x$ and $y$ intercepts.

$b$ Find the equations of the vertical and horizontal asymptotes.

$c$ Find the inverse function $f^{- 1}(x)$.

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 $x = 2$.

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 $q$.

Consider the functions $f(x) = x^2 - 4$ and $g(x) = \frac{p}{x}$, where $p$ is a non-zero real constant.

(a) Write down an expression for $(f \circ g)(x)$.

(b) Given that $(f \circ g)(2) = 5$, find the possible values of $p$.

Consider the functions $f(x) = \sin x - \cos x$ and $h(x) = x + \frac{\pi}{4}$ for $x \in \mathbb{R}$.

(a) Find an expression for $(f \circ h)(x)$.

(b) Hence, solve the equation $(f \circ h)(x) = 1$ for $0 \le x \le 2\pi$.

Consider the function f defined by $f(x) = 50e^{-0.2x}$ for $x \in \mathbb{R}^+$.

The graph of f and the line $y = x$ intersect at point P.

Find the x-coordinate of P.

The line L has a gradient of -2 and is a tangent to the graph of f at the point Q.

Find the exact coordinates of Q.

Show that the equation of L is $y = -2x + 10 \ln 5 + 10$.

The shaded region A is enclosed by the graph of f and the lines $y = x$ and L.

Find the x-coordinate of the point where L intersects the line $y = x$.

Hence, find the area of A.

The line L is tangent to the graphs of both f and the inverse function $f^{-1}$.

Find the shaded area enclosed by the graphs of f and $f^{-1}$ and the line L.

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}$, $x \neq -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

A company models the efficiency of a new production line by the function $f(x) = \frac{2x+3}{x+2}$, where $f(x)$ represents the output rate (in units per hour) after $x$ hours of operation. For mathematical analysis, we consider the function over its natural domain $x \in \mathbb{R}$, $x \ne -2$.

For the graph of f,

write down the equation of the vertical asymptote;

find the equation of the horizontal asymptote.

Find $f^{-1}(x)$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of f in the y-axis followed by a reflection in the x-axis.

The graphs of f and $f^{-1}$ intersect at $x = p$ and $x = q$, where $p < q$.

Find the value of p and the value of q.

Hence, find the area enclosed by the graph of f and the graph of $f^{-1}$.

Consider the functions $f(x) = e^{2x}$ and $g(x) = x+c$, where $c$ is a real constant.

(a) Write down an expression for $(f \circ g)(x)$.

(b) Given that $(f \circ g)(\ln 3) = 36$, find the value of $c$.

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 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.

Consider the functions $f(x) = \sin x + \sqrt{3}\cos x$ and $g(x) = 2x$.

(a) Find $(f \circ g)(x)$.

(b) Solve the equation $(f \circ g)(x) = 2\sin(2x)$ for $0 \leq x \leq \pi$.

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

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}, x \ne -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

Consider the functions $p$, $q$ and $r$ defined by $p(x) = x^2 + 3$, $q(x) = 2x + 1$ and $r(x) = 7$.

(a) Write down the value of $p(2)$.

(b) Write down the value of $p(-4)$.

(c) Write down the value of $q(8)$.

(d) Find the value of $p(-1) + r(5)$.

(e) Find the value of $3q(1) - p(0)$.

(f) Find the value of $r(-2) \times q(-3)$.

(g) Find the value of $q^{-1}(6)$.

(h) Find an expression for the composite function $(p \circ q)(x)$, giving your answer in the form $ax^2+bx+c$.

(i) Find an expression for the composite function $(p \circ q^{-1})(x)$.

The function $f$ is defined by $f(x) = \frac{3x - 1}{x + 2}$ for $x \in \mathbb{R}, x \neq -2$.

(a) Find the range of $f$.

(b) Find an expression for the inverse function, $f^{-1}(x)$.

(c) Find an expression for $f(f(x) )$.

(d) Solve the equation $f(x) = x$.

(e) Hence, or otherwise, state the number of solutions to the equation $f(x) = f^{-1}(x)$ and justify your answer.

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 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 $k$.