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Worksheet
Functions I: Domain, Range & Inverse
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
State the domain and range of the following functions:
(a)\(f(x)=x^2+3,\quad x\in\mathbb{R}\)
(b)\(f(x)=\dfrac{1}{x-2},\quad x\in\mathbb{R},\;x eq2\)
(c)\(f(x)=\sqrt{x+4},\quad x\geq-4\)
(d)\(f(x)=\sin x,\quad x\in\mathbb{R}\)
(a)
Answer
Domain: \(\mathbb{R}\); Range: \(f(x)\geq3\)
(b)
Answer
Domain: \(x eq2\); Range: \(f(x) eq0\)
(c)
Answer
Domain: \(x\geq-4\); Range: \(f(x)\geq0\)
(d)
Answer
Domain: \(\mathbb{R}\); Range: \(-1\leq f(x)\leq1\)
Question 2
For each function find \(f^{-1}(x)\) and state its domain:
(a)\(f(x)=3x-5\)
(b)\(f(x)=x^2-1,\quad x\geq0\)
(c)\(f(x)=\dfrac{2x+1}{x-3},\quad x eq3\)
(a)
Swap and rearrange
\(y=3x-5\Rightarrow x=\dfrac{y+5}{3}\)
Answer
\(f^{-1}(x)=\dfrac{x+5}{3},\quad x\in\mathbb{R}\)
(b)
Swap
\(y=x^2-1\Rightarrow x=\sqrt{y+1}\)
Answer
\(f^{-1}(x)=\sqrt{x+1},\quad x\geq-1\)
(c)
Swap and rearrange
\(y(x-3)=2x+1\Rightarrow x(y-2)=3y+1\)
Answer
\(f^{-1}(x)=\dfrac{3x+1}{x-2},\quad x eq2\)
Question 3
Given \(f(x)=2x+3\) and \(g(x)=x^2-1\), find:
(a)\(f(g(x))\)
(b)\(g(f(x))\)
(c)\(f(f(x))\)
(d)\(g(g(2))\)
(e)\(f^{-1}(g(x))\)
(a)
Answer
\(2x^2-2+3=2x^2+1\)
(b)
Answer
\((2x+3)^2-1=4x^2+12x+8\)
(c)
Answer
\(2(2x+3)+3=4x+9\)
(d)
Step by step
\(g(2)=3;\;g(3)=8\)
Answer
\(8\)
(e)
\(f^{-1}(x)= frac{x-3}{2}\)
\(f^{-1}(g(x))=\dfrac{x^2-1-3}{2}\)
Answer
\(\dfrac{x^2-4}{2}\)
Question 4
Solve \(f(f(x))=x\) for \(f(x)=\dfrac{x+2}{x-1},\;x eq1\).
Compute \(f(f(x))\)
\(f(f(x))=\dfrac{ frac{x+2}{x-1}+2}{ frac{x+2}{x-1}-1}=\dfrac{x+2+2(x-1)}{x+2-(x-1)}=\dfrac{3x}{3}=x\)
Answer
\(f(f(x))=x\) for all \(x eq1\) — this function is its own inverse.
Question 5
The function \(f(x)=ax+b\) satisfies \(f(2)=7\) and \(f^{-1}(3)=1\). Find \(a\) and \(b\).
Two equations
\(2a+b=7\quad(1)\); \(f^{-1}(3)=1\Rightarrow f(1)=3\Rightarrow a+b=3\quad(2)\)
Solve
\((1)-(2):a=4\Rightarrow b=-1\)
Answer
\(a=4,\;b=-1\)
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