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

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

[2]

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

[2]

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

[3]

Question 2

SLPaper 1

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

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

Write down the equation of the vertical asymptote of the graph of $f$ .

[1]

Find the equation of the horizontal asymptote of the graph of $f$ .

[2]

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

[4]

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

[4]

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

[2]

Show that the line $y=-x+1$ is a line of symmetry for the points of intersection of the graphs of $f$ and $f^{-1}$ .

[3]

Question 3

SLPaper 1

The functions $f$ and $g$ are defined for $x \in R$ by $f(x)=2 x+3$ and $g(x)=-x+k$ , where $k \in R$ .

Find the range of $f$ .

[2]

Given that $(g \circ f)(x) \leq 0$ for all $x \in R$ , determine the set of possible values for $k$ .

[3]

Question 4

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 value of $a$ .

[3]

Question 5

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 $(h \circ k)(x)$ with range $y\in\R,y\not=\frac{1}{4}$

[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 $\int\! f(x)dx$ .

[5]

Question 6

SLPaper 1

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

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 7

SLPaper 2

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

Given that $k=2$ , find $f^{-1}(x)$ .

[4]

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

[4]

Question 8

SLPaper 1

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

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

Write down the equation of the vertical asymptote of the graph of $f$ .

[1]

Find the equation of the horizontal asymptote of the graph of $f$ .

[2]

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

[4]

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

[4]

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

[2]

Question 9

SLPaper 1

Consider the function $f(x)=b^{x}$ , where $x, b \in \mathbb{R}$ , $x>0$ , $b>1$ . The graph of $f$ contains the point $\left(\frac{1}{2}, 3\right)$ . The terms of an arithmetic sequence are formed by applying the inverse of $f(x)$ to the terms of a geometric sequence. Specifically, consider the arithmetic sequence $\log _{9} 27, \log _{9} p, \log _{9} q, \log _{9} 243$ , where $p>1$ and $q>1$ .

Show that $b=9$ .

[2]

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

[1]

Find the value of $f^{-1}(27)$ .

[3]

Show that $27, p, q, 243$ are four consecutive terms in a geometric sequence.

[4]

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

[4]

Question 10

SLPaper 2

Let $f(x)=e^{kx}$ and $g(x)=\ln(x+2)$ for $x>-2$ and $k>0$ .

It is given that $(f\circ g)(3)=4$ .

Determine $k$ correct to 3 significant figures.

[3]

Find explicit expressions for $(f\circ g)(x)$ and $(g\circ f)(x)$ . State the maximal domains for each composite.

[4]

Using your value of $k$ , solve $(f\circ g)(x)=(g\circ f)(x)$ for $x\ge0$ to three decimal places, and justify that there is exactly one solution on $x\ge0$ .

[7]

Find $(f\circ g)^{-1}(y)$ and state its domain and range.

[4]

SL 2.5—Composite functions, identity, finding inverse Questionbank