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SL 2.5—Composite functions, identity, finding inverse - IB Questionbank | RevisionDojo

Practice SL 2.5—Composite functions, identity, finding inverse with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Let $f(x)=2x-3$ and $g(x)=x^2$ , for $x \in \mathbb{R}$ .

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

[2]

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

[2]

Given that $f^{-1}(b)=g(b)$ , find the values of $b$ .

[3]

Question 2

HLPaper 1

Let $h(x) = \frac{1}{x^2}$ and $k(x) = \ln(x)$ .

Find the composite function $(h \circ k)(x)$ .

[1]

Determine the domain of the composite function $(h \circ k)(x)$ , given that $(h \circ k)(x) \neq \frac{1}{4}$ .

[5]

Evaluate the limit $\lim_{x \to 1} (h \circ k)(x)$ .

[2]

Now consider the function $f(x)=k(x)\sqrt{h(x)}$ . Hence, find $\int f(x) \, \mathrm{d}x$ .

[5]

Question 3

SLPaper 1

Let $f(x)=3 x-1$ and $g(x)=\sqrt{x}$ , for $x \geq 0$ .

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

[2]

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

[2]

Given that $f(a)=g^{-1}(a)$ , find the values of $a$ .

[3]

Question 4

SLPaper 2

Let $f(x) = x - 8$ , $g(x) = x^4 - 3$ , and $h(x) = f(g(x))$ .

Find $h(x)$ .

[2]

Let $D$ be a point on the graph of $h$ . The tangent to the graph of $h$ at $D$ is parallel to the graph of $f$ . Find the $x$ -coordinate of $D$ .

[5]

Question 5

SLPaper 1

Consider the function $f(x)=k x^{2}$ , where $x, k \in \mathbb{R}, x>0, k>0$ .

The graph of $f$ contains the point $(\sqrt{2}, 8)$ .

Consider the arithmetic sequence $\log _{4} 16, \log _{4} p, \log _{4} q, \log _{4} 256$ , where $p>1$ and $q>1$ .

Show that $k=4$ .

[2]

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

[1]

Find the value of $f^{-1}(2)$ .

[3]

Show that $16, p, q, 256$ are four consecutive terms in a geometric sequence.

[4]

Find the value of $p$ and the value of $q$ .

[4]

Question 6

SL & HLPaper 1

Consider the function $h(x) = x^2 + 2x + 1$ for $x \geq -1$ .

Find the inverse of the function $h(x)$ .

[4]

Graph the function $h(x) = x^2 + 2x + 1$ and its inverse on the same set of axes.

[3]

Question 7

SLPaper 2

Let $f(x)=\ln \left(1+e^{x}\right)$ and $g(x)=k x+c$ , for $x \in \mathbb{R}$ , where $k$ and $c$ are constants.

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

[2]

Given that $\lim _{x \rightarrow-\infty}(g \circ f)(x)=-2$ , find the value of $c$ .

[3]

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

[4]

With $k=2$ and the value of $c$ from Part 2, solve the equation $(g \circ f)(x)= x$ for $x$ .

[4]

Question 8

SLPaper 2

The functions $h$ and $j$ are defined for $x \in \mathbb{R}$ by $h(x) = 6x^2 - 12x + 1$ and $j(x) = -x + d$ , where $d \in \mathbb{R}$ .

Find the range of $h$ .

[2]

Given that $(j \circ h)(x) \le 0$ for all $x \in \mathbb{R}$ , determine the set of possible values for $d$ .

[4]

Question 9

HLPaper 2

Consider the function $f(x) = \frac{2x^2 - 3x + 1}{x^2 - 4}$ .

Find the domain of the function $f(x)$ .

[4]

Draw the graph of $f(x)$ clearly indicating any asymptotes and intercepts of the function.

[3]

Find the inverse of the function $f(x)$ over the restricted domain $[1, 2) \cup (2, 3]$ if it exists.

[6]

Question 10

SLPaper 1

Consider the function $f(x) = 2x + 3$ .

Find the inverse of the function $f(x)$ .

[2]

Graph the function $f(x) = 2x + 3$ and its inverse on the same set of axes.

[2]

SL 2.5—Composite functions, identity, finding inverse Questionbank