A-Level Maths | Functions
Question 1
Find the range for each of the following functions.
a) $f(x)=x^2+1,\ x\in\mathbb{R},\ x>0$
b) $g(x)=x^2-8x+13,\ x\in\mathbb{R},\ x\geq0$
c) $h(x)=x^2+2x+2,\ x\in\mathbb{R},\ -5\leq x<-2$
d) $f(x)=x^2-6x+6,\ x\in\mathbb{R}$
e) $g(x)=x^2+8x+12,\ x\in\mathbb{R},\ -3\leq x\leq0$
f) $h(x)=x^2-10x+26,\ x\in\mathbb{R},\ x\geq0$
g) $f(x)=x^2-4x+3,\ x\in\mathbb{R},\ x>2$
h) $g(x)=x^2+4x+2,\ x\in\mathbb{R},\ x\geq0$
i) $h(x)=\dfrac{1}{x-2},\ x\in\mathbb{R},\ x>2$
j) $g(x)=\dfrac{2}{x+3},\ x\in\mathbb{R},\ x\geq1$
k) $h(x)=\dfrac{1}{x-1}+2,\ x\in\mathbb{R},\ x>2$
l) $h(x)=\dfrac{1}{x-2},\ x\in\mathbb{R},\ x\neq2$
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a) $f(x)\in\mathbb{R},\ f(x)>1$
b) $g(x)\in\mathbb{R},\ g(x)\geq-3$
c) $h(x)\in\mathbb{R},\ 2<h(x)\leq17$
d) $f(x)\in\mathbb{R},\ f(x)\geq-3$
e) $g(x)\in\mathbb{R},\ -3\leq g(x)\leq12$
f) $h(x)\in\mathbb{R},\ h(x)\geq1$
g) $f(x)\in\mathbb{R},\ f(x)>-1$
h) $g(x)\in\mathbb{R},\ g(x)\geq2$
i) $f(x)\in\mathbb{R},\ f(x)>0$
j) $g(x)\in\mathbb{R},\ 0<g(x)\leq\dfrac{1}{2}$
k) $h(x)\in\mathbb{R},\ 2<h(x)<3$
l) $h(x)\in\mathbb{R},\ h(x)\neq0$
Question 2
Find $fg(x)$ and $gf(x)$ for each of the following, simplifying your answers.
a) $f(x)=2x+1,\ x\in\mathbb{R}$ and $g(x)=x^2-1,\ x\in\mathbb{R}$
b) $f(x)=4-3x,\ x\in\mathbb{R}$ and $g(x)=\sqrt{x},\ x\in\mathbb{R},\ x\geq0$
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a) $fg(x)=2x^2-1$, $gf(x)=4x^2+4x$
b) $fg(x)=4-3\sqrt{x}$, $gf(x)=\sqrt{4-3x}$
Question 3
For each of the following functions find an expression for its inverse.
a) $f(x)=4x-1,\ x\in\mathbb{R}$
b) $g(x)=1+\sqrt{x},\ x\in\mathbb{R},\ x\geq0$
c) $h(x)=1-\sqrt{x-5},\ x\in\mathbb{R},\ x\geq5$
d) $f(x)=5-2x,\ x\in\mathbb{R}$
e) $g(x)=\dfrac{3}{x}-2,\ x\in\mathbb{R},\ x\neq0$
f) $h(x)=\sqrt{\dfrac{x}{2}}-1,\ x\in\mathbb{R},\ x\geq2$
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a) $f^{-1}(x)=\dfrac{x+1}{4}$
b) $g^{-1}(x)=(x-1)^2$
c) $h^{-1}(x)=5+(1-x)^2$
d) $f^{-1}(x)=\dfrac{5-x}{2}$
e) $g^{-1}(x)=\dfrac{3}{x+2}$
f) $h^{-1}(x)=2x^2+2$
Question 4
For each of the following functions, find an expression for $f^{-1}(x)$ and state its domain and range.
a) $f(x)=(x-1)^2,\ x\in\mathbb{R},\ x\geq1$
b) $f(x)=\sqrt{x+4},\ x\in\mathbb{R},\ 0\leq x<5$
c) $f(x)=e^{2x}-1,\ x\in\mathbb{R}$
d) $f(x)=2+\dfrac{1}{x+1},\ x\in\mathbb{R},\ x\geq0$. Also sketch $f(x)$ and $f^{-1}(x)$ on the same diagram.
e) $f(x)=4-\dfrac{1}{x-1},\ x\in\mathbb{R},\ x>1$
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a) $f^{-1}(x)=1+\sqrt{x}$; domain $x\geq0$, range $f^{-1}(x)\geq1$
b) $f^{-1}(x)=x^2-4$; domain $2\leq x<3$, range $0\leq f^{-1}(x)<5$
c) $f^{-1}(x)=\dfrac{1}{2}\ln(x+1)$; domain $x>-1$, range $f^{-1}(x)\in\mathbb{R}$
d) $f^{-1}(x)=\dfrac{3-x}{x-2}$; domain $2<x\leq3$, range $f^{-1}(x)\geq0$
e) $f^{-1}(x)=\dfrac{x-5}{x-4}$; domain $x<4$, range $f^{-1}(x)>1$
Question 5
Solve each equation.
a) $|x-5|=3$
b) $|x+1|=15$
c) $|2x-7|=4$
d) $|x-2|=|x+4|$
e) $|x-5|=|7-x|$
f) $|2x+1|=|9-2x|$
g) $|x+3|=|2x|$
h) $|4x-1|=|2-x|$
i) $|3x-4|=|2x+3|$
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a) $x=2$ or $x=8$
b) $x=-16$ or $x=14$
c) $x=\dfrac{3}{2}$ or $x=\dfrac{11}{2}$
d) $x=-1$
e) $x=6$
f) $x=2$
g) $x=-1$ or $x=3$
h) $x=-\dfrac{1}{3}$ or $x=\dfrac{3}{5}$
i) $x=\dfrac{1}{5}$ or $x=7$
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