Functions: 43 IB Maths AA HL practice questions

The function $f$ is defined by $f(x) = \frac{1}{x}$, $x \neq 0$.

The graph of the function $y = g(x)$ is obtained by applying the following transformations to the graph of $y = f(x)$:

a translation by the vector $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$;

a translation by the vector $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

(a) Find an expression for $g(x)$.[2]

(b) State the equations of the asymptotes of the graph of $g$.[2]

The quadratic function $f(x) = p + qx - x^2$ has a maximum value of 5 when $x = 3$.

(a) Find the value of $p$ and the value of $q$.[4]

(b) The graph of $f(x)$ is translated 3 units in the positive direction parallel to the $x$-axis. Determine the equation of the new graph.[2]

Consider the function defined by $f(x) = x^3 - 3x^2 + 4$.

(a) Determine the values of $x$ for which $f(x)$ is a decreasing function.[4]

There is a point of inflexion, P, on the curve $y = f(x)$.

(b) Find the coordinates of P.[3]

The function $f$ is defined by $f(x) = \frac{7x + 7}{2x - 4}$ for $x \in \mathbb{R}, x \neq 2$.

(a) Find the zero of $f(x)$.[2]

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

(i) the vertical asymptote;[2]

(ii) the horizontal asymptote.

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

(a) Express the quadratic $3x^2 - 6x + 5$ in the form $a(x+b)^2 + c$, where $a, b, c \in \mathbb{Z}$.[3]

(b) Describe a sequence of transformations that transforms the graph of $y = x^2$ to the graph of $y = 3x^2 - 6x + 5$.[3]

The function $f$ is defined by

$$f(x) = \begin{cases} 1-2x, & x \leq 2 \\ \frac{3}{4}(x-2)^2-3, & x > 2 \end{cases}$$

(a) Determine whether or not $f$ is continuous.[2]

The graph of the function $g$ is obtained by applying the following transformations to the graph of $f$:

a reflection in the $y$-axis followed by a translation by the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$.

(b) Find $g(x)$.[4]

The function $f$ is defined by $f(x) = 2x^3 + 5$, $-2 \le x \le 2$.

(a) Write down the range of $f$.[2]

(b) Find an expression for $f^{-1}(x)$.[2]

(c) Write down the domain and range of $f^{-1}$.[2]

Consider the functions $f(x) = x - 3$ and $g(x) = x^2 + k^2$, where $k$ is a real constant.

(a) Write down an expression for $(g \circ f)(x)$.[2]

(b) Given that $(g \circ f)(2) = 10$, find the possible values of $k$.[3]

Consider the function $f(x) = a\tan(2x) + b$, where $x \neq \frac{(2n+1)\pi}{4}$, $n \in \mathbb{Z}$ and $a, b \in \mathbb{R}$.

(a) Write down the period of $f$.[1]

The graph of $y = f(x)$ passes through the points $\left(\frac{\pi}{12}, 5\right)$ and $\left(\frac{\pi}{3}, 7\right)$.

(b) Find the value of $a$ and the value of $b$.[3]

The functions $f$ and $g$ are defined such that $f(x) = \frac{x + 3}{4}$ and $g(x) = 8x + 5$.

(a) Show that $(g \circ f)(x) = 2x + 11$.[2]

(b) Given that $(g \circ f)^{-1}(a) = 4$, find the value of $a$.[3]

The functions $f$ and $g$ are defined by $f(x) = ax^2 + bx + c, x \in \mathbb{R}$ and $g(x) = p\sin x + qx + r, x \in \mathbb{R}$ where $a, b, c, p, q, r$ are real constants.

(a) Given that $f$ is an even function, show that $b = 0$.[2]

(b) Given that $g$ is an odd function, find the value of $r$.[2]

The function $h$ is both odd and even, with domain $\mathbb{R}$.

(c) Find $h(x)$.[2]

Consider the functions $f(x) = x - 3$ and $g(x) = x^2 + k^2$, where $k$ is a real constant.

(a) Write down an expression for $(g \circ f)(x)$.[2]

(b) Given that $(g \circ f)(2) = 10$, find the possible values of $k$.[3]

The function $f$ is defined as $f(x) = ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$.

Hayley conjectures that $\frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f'(x_2) + f'(x_1)}{2}$, $x_1 \neq x_2$.

Show that Hayley's conjecture is correct.[6]

The function $f$ is defined by $f(x) = \sec x + 2$, $0 \le x < \frac{\pi}{2}$.

(a) Write down the range of $f$.[1]

(b) Find $f^{-1}(x)$, stating its domain.[4]

A function $f$ is defined by $f(x) = x^3 + e^x + 1, x \in \mathbb{R}$. By considering $f'(x)$ determine whether $f$ is a one-to-one or a many-to-one function.[4]