AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction Questionbank

Practice AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction 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

HLPaper 1

Consider the function $g(x)=\ln |x+3|, x \in \mathbb{R}, x \neq-3$ .

State the range of $g$ .

[1]

Find the coordinates of the point where the graph of $y=g(x)$ intersects the y-axis. marks

[2]

Question 2

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x-3},\, x\ne3.$

Express $f(x)=x+p+3+\dfrac{q+3p+9}{x-3}$ and state the slant asymptote.

[3]

Prove $f$ is neither even nor odd for any $p,q$ .

[4]

Parameters for exactly one $x$ -intercept; give its coordinate.

[7]

For $q+3p+9\ne0$ and $\Delta>0$ , solve $f(x)\ge x+p+3$ .

[6]

Question 3

HLPaper 2

Consider the function $f(x)=\frac{1}{x^{2}-6 x+5}, x \in \mathbb{R}, x \neq 1,5$ , and the function $g(x)=\frac{1}{x^{2}-6 x+5}$ , $x>5$ . Let $h(x)=\arcsin \left(\frac{2 x}{x+2}\right), x>-2$ .

Sketch the graph of $y=f(x)$ , indicating all asymptotes with their equations, the y-intercept, and any local maximum or minimum points.

[8]

Show that $g^{-1}(x)=3+\sqrt{\frac{4 x+1}{x}}$ .

[5]

State the domain and range of $g^{-1}$ .

[2]

Determine whether $f$ is an even or odd function, justifying your answer.

[3]

Solve $(h \circ g)(a)=\frac{\pi}{3}$ , giving the answer in the form $p+q \sqrt{r}$ , where $p, q, r \in \mathbb{Z}^{+}$ .

[5]

Hence, or otherwise, find the x -coordinates of the points where the graphs of $y=f(x)$ and $y=g^{-1}(x)$ intersect for $x>5$ , and determine the intervals where $f(x)>g^{-1}(x)$ .

[3]

Question 4

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x+3},\,x\ne-3.$

Divide to obtain $f(x)=Q(x)+\dfrac{R(x)}{x+3}$ and the slant asymptote.

[3]

Show $f$ is neither even nor odd for any $p,q$ .

[4]

Find all $(p,q)$ for exactly one $x$ -intercept; give it.

[7]

For $q-3p+9\ne0$ and $\Delta>0$ , solve $f(x)\ge x+p-3$ .

[6]

Question 5

HLPaper 2

Given $f(x)=\frac{ax+2}{-x+a},\quad a\in\mathbb R,\ x\ne a.$

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

[3]

Determine when $f$ is self-inverse.

[2]

For $a=0$ , find fixed points and solve $f(x)\ge x$ .

[4]

For $a=1$ , solve $f(x)=\sqrt{x+5}$ numerically to 3 s.f., stating the number of solutions.

[3]

Question 6

HLPaper 2

Define $f(x)=\frac{x-1}{kx+2},\quad k\in\mathbb R,\ x\ne -\frac{2}{k}\ (k\ne0).$

Compute $f^{-1}(x)$ and its domain.

[3]

For which $k$ is $f$ self-inverse?

[2]

Let $g_k(x)=\dfrac{kx+1}{x-k}$ . Prove self-inverse and find fixed points.

[3]

For $k>0$ , solve $g_k(x)\ge x$ . For $k=2$ , solve $g_k(x)=\ln(1+x)$ to 3 s.f., noting the number of roots.

[6]

Question 7

HLPaper 2

Given $f(x)=\frac{mx+n}{px-q},\quad mq+np\ne0,\quad x\ne \frac{q}{p}.$

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

[3]

Show $f$ is self-inverse iff $m-q=0$ .

[4]

Let $g_s(x)=\dfrac{s x+1}{x-s}$ with $s\ne0$ . Prove self-inverse and find fixed points.

[3]

For $s>0$ , solve $g_s(x)\ge x$ . For $s=1$ , solve $g_s(x)=e^{-x}$ to 3 s.f., commenting on uniqueness.

[7]

Question 8

HLPaper 2

Consider the function $g(x)=\ln (2 x+4), x>-2$ .

State the range of $g$ .

[1]

Describe the transformation that maps $y=\ln x$ to $y=g(x)$ .

[2]

Find the coordinates of the point where the graph of $y=g(x)$ intersects the line $y=\ln 2$ . marks

[3]

Question 9

HLPaper 2

Consider the function $f(x)=\ln x-\frac{2}{x-2}, 0<x<2$ . Draw a set of axes showing $x$ and $y$ values between -2 and 4 . On these axes.

Hence, or otherwise, find the coordinates of the point of inflection on the graph of $y=f(x)$ .

[3]

Sketch the graph of $y=f(x)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.

[5]

Sketch the graph of $y=f^{-1}(x)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.

[5]

Question 10

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x+6},\, x\ne-6.$

Divide and give the oblique asymptote.

[3]

Show $f$ is neither even nor odd.

[4]

Find all $(p,q)$ for exactly one $x$ -intercept; include its coordinate.

[7]

For $q-6p+36\ne0$ and $\Delta>0$ , solve $f(x)\ge x+p-6$ .

[6]