IB Maths AA HL Topic 2: Functions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 1, no calculatorHard6 marks

A non-constant quadratic function $f$ satisfies $f(x)=f(4-x)$ for every real $x$ It is also known that $f(4)=-2$ and $f(2)=6$

(a)

Determine $f(0)$

[1]

(b)

Determine $f(x)$ in vertex form.

[2]

(c)

Hence determine all real values of $m$ for which the line $y=mx$ intersects the graph of $y=f(x)$ at two distinct points.

[3]

Question 2

Paper 1, no calculatorHard6 marks

Figure 1 shows the graph of a quadratic function $f$ The marked points lie exactly on grid intersections.

(a)

Determine $f(x)$ in factorized form.

[2]

(b)

A quadratic function $g$ has the same $x$ -intercepts as $f$ The $y$ -coordinate of the vertex of $g$ is equal to the $y$ -intercept of $f$

Determine $g(x)$ in the form $a(x-h)^2+k$

[4]

Question 3

Paper 1, no calculatorEasy6 marks

Consider the functions

f(x)=\frac1x,\quad x\ne 0, \qquad g(x)=\frac{2x+3}{x-1},\quad x\ne 1.

(a)

Write down $f^{-1}(x)$

[1]

(b)

State the equations of the vertical and horizontal asymptotes of the graph of $y=g(x)$

[2]

(c)

Sketch the graph of $y=g(x)$ showing both asymptotes and all intercepts with the coordinate axes.

[3]

Question 4

Paper 1, no calculatorHard6 marks

Let $f:\mathbb R\to\mathbb R$ be odd and have period $T>0$ so that $f(x+T)=f(x)$ for all $x\in\mathbb R$

(a)

Explain why $f$ cannot have an inverse function when its domain is all of $\mathbb R$

[2]

(b)

Let $I\subseteq\mathbb R$ be symmetric about $0$ and suppose the restriction of $f$ to $I$ is one-to-one. Let $g$ be the inverse of this restriction, with domain $f(I)$ Show that $g$ is odd.

[4]

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