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IB Maths AA: Functions — Practice Questions & Answers
A complete tour of IB AA Topic 2: function notation, domain and range, composite and inverse functions, transformations of graphs, quadratics and the discriminant, polynomial division with the factor and remainder theorems, rational functions and their asymptotes, and the exponential and logarithmic families. Covers both SL and HL with the HL-only extensions flagged throughout.
50 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
Given $f(x)=3x-5$, what is the value of $f(4)$?
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Why Substitute $x=4$ to get $f(4)=3(4)-5=12-5=7$.
2
Multiple choice · Easy
What is the domain of the function $f(x)=\sqrt{x-2}$?
$x\ge 2$
$x\le 2$
$x\ge 0$
$x\ge -2$
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Why The expression under the square root must be non-negative, so $x-2\ge 0$, giving $x\ge 2$.
3
Multiple choice · Easy
The graph of $y=x^{2}$ is translated $3$ units upward. What is the equation of the new graph?
$y=x^{2}+3$
$y=x^{2}-3$
$y=(x+3)^{2}$
$y=(x-3)^{2}$
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Why A vertical translation up by $3$ adds $3$ to the function, giving $y=x^{2}+3$.
4
Multiple choice · Easy
Which transformation maps $y=f(x)$ to $y=-f(x)$?
Reflection in the $x$-axis
Reflection in the $y$-axis
Translation downward
Vertical stretch
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Why Multiplying the output by $-1$ reflects the graph in the $x$-axis.
5
Multiple choice · Easy
What is the $y$-intercept of the function $f(x)=2^{x}$?
$1$
$0$
$2$
$\frac{1}{2}$
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Why At $x=0$, $f(0)=2^{0}=1$, so the $y$-intercept is $1$.
6
Multiple choice · Easy
For the quadratic $f(x)=(x-3)^{2}+4$, what are the coordinates of the vertex?
$(3,4)$
$(-3,4)$
$(3,-4)$
$(-3,-4)$
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Why In vertex form $a(x-h)^{2}+k$, the vertex is $(h,k)=(3,4)$.
7
Multiple choice · Easy
What is the horizontal asymptote of $f(x)=\frac{1}{x}+2$?
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Why As $x$ tends to infinity, $\frac{1}{x}$ tends to $0$, so $y$ approaches $2$.
8
Multiple choice · Easy
Given $f(x)=x+1$ and $g(x)=2x$, what is $(f\circ g)(x)$?
$2x+1$
$2x+2$
$2(x+1)$
$x+2$
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Why $(f\circ g)(x)=f(g(x))=f(2x)=2x+1$.
9
Multiple choice · Medium
The function $f(x)=4x-7$ has inverse $f^{-1}(x)$. What is $f^{-1}(x)$?
$\frac{x+7}{4}$
$\frac{x-7}{4}$
$\frac{1}{4x-7}$
$4x+7$
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Why Let $y=4x-7$, swap and solve: $x=4y-7$ gives $y=\frac{x+7}{4}$.
10
Multiple choice · Medium
For what value of $k$ does the equation $x^{2}-6x+k=0$ have exactly one real root?
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Why One root means the discriminant is zero: $(-6)^{2}-4(1)(k)=0$, so $36-4k=0$ and $k=9$.
Key terms in Functions
Function: A rule that assigns to each input exactly one output.
Mapping: A relationship that associates elements of one set with elements of another.
Domain: The set of all allowed input values of a function.
Range: The set of all output values a function can produce.
Codomain: The set within which the outputs of a function are required to lie.
Independent variable: The input variable of a function, usually $x$.
Dependent variable: The output variable of a function, usually $y$, whose value depends on the input.
Argument of a function: The input value placed into a function, such as $x$ in $f(x)$.
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