Functions: 57 IB Maths AA SL practice questions

Let $f(x) = 3x$, $g(x) = 2x - 5$ and $h(x) = (f \circ g)(x)$.

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

(b) Find $h^{-1}(x)$.[3]

Two functions, $f$ and $g$, are defined in the following table.

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

(b) Find the value of $(g \circ f)(1)$.[2]

(c) Find the value of $g^{-1}(-2)$.[2]

Let $f(x) = 8x + 3$ and $g(x) = 4x$, for $x \in \mathbb{R}$.

(a) Write down $g(2)$.[1]

(b) Find $(f \circ g)(x)$.[2]

(c) Find $f^{-1}(x)$.[2]

Let $f(x) = 3x^2$. The graph of $f$ is translated 1 unit to the right and 2 units down. The graph of $g$ is the image of the graph of $f$ after this translation.

(a) Write down the coordinates of the vertex of the graph of $g$. [2 marks]

(b) Express $g$ in the form $g(x) = 3(x - p)^2 + q$. [2 marks]

The graph of $h$ is the reflection of the graph of $g$ in the $x$-axis.

(c) Write down the coordinates of the vertex of the graph of $h$. [2 marks]

Let $f(x) = \sqrt{x+2}$ for $x \geq -2$ and $g(x) = 3x - 7$ for $x \in \mathbb{R}$.

(a) Write down $f(14)$.[1]

(b) Find $(g \circ f)(14)$.[2]

(c) Find $g^{-1}(x)$.[3]

The following table shows values of $f(x)$ and $g(x)$ for different values of $x$.

Both $f$ and $g$ are one-to-one functions.

(a) Find $g(0)$.[1]

(b) Find $(f \circ g)(0)$.[2]

(c) Find the value of $x$ such that $f(x) = 0$.[2]

Let $f(x) = 2x - 1$ and $g(x) = 3x^2 + 2$.

(a) Find $f^{-1}(x)$.[3]

(b) Find $(f \circ g)(1)$.[3]

Let $f(x) = 2x + 4$ and $g(x) = 7x^2$.

(a) Find $f^{-1}(x)$.[3]

(b) Find $(f \circ g)(x)$.[2]

(c) Find $(f \circ g)(3.5)$.[2]

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]

Let $f(x) = 3 \ln x$ and $g(x) = \ln 5x^3$.

(a) Express $g(x)$ in the form $f(x) + \ln a$, where $a \in \mathbb{Z}^+$.[4]

(b) The graph of $g$ is a transformation of the graph of $f$. Give a full geometric description of this transformation.[3]

Let $f(x) = x^2 + 2x + 1$ and $g(x) = x - 5$, for $x \in \mathbb{R}$.

(a) Find $f(8)$.[2]

(b) Find $(g \circ f)(x)$.[2]

(c) Solve $(g \circ f)(x) = 0$.[3]

Let $f(x) = \frac{6x-1}{2x+3}$, for $x \neq -\frac{3}{2}$.

(a) For the graph of $f$,

(i) find the $y$-intercept;[2]

(ii) find the equation of the vertical asymptote;[1]

(iii) find the equation of the horizontal asymptote.[2]

(b) Hence or otherwise, write down $\lim_{x \to \infty} \left(\frac{6x-1}{2x+3}\right)$.[2]

Consider the function $f(x) = \frac{3x + 1}{x - 2}$, $x \neq 2$.

(a)(i) write down the equation of the vertical asymptote;[2]

(a)(ii) find the equation of the horizontal asymptote.[1]

Let $g(x) = x^2 + 4$, $x \in \mathbb{R}$.

(b) Find $(f \circ g)(1)$.[3]

Let $f$ and $g$ be functions such that $g(x) = 2f(x+1) + 5$.

(a) The graph of $f$ is mapped to the graph of $g$ under the following transformations: vertical stretch by a factor of $k$, followed by a translation $\begin{pmatrix} p \\ q \end{pmatrix}$. Write down the value of[3]

(i) $k$;

(ii) $p$;

(iii) $q$.

(b) Let $h(x) = -g(3x)$. The point $A(6, 5)$ on the graph of $g$ is mapped to the point $A'$ on the graph of $h$. Find $A'$.[3]

Consider the functions $f(x) = x + 2$ 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 $(g \circ f)(4) = 11$, find the possible values of $k$.[3]