Solutions To Mathematics Textbooks/Basic Mathematics/Chapter 13

Chapter 13

Section 1

1

$f(x)=\frac{1}{x}$

$f\left(\frac{3}{4}\right)=$ $\frac{1}{\frac{3}{4}}=$ $1*{\left(\frac{3}{4}\right)}^{-1}=\frac{4}{3}$

$f(-\frac{2}{3})=$ $\frac{1}{-\frac{2}{3}}=$ $1*{\left(\frac{-2}{3}\right)}^{-1}=$ $\frac{3}{-2}$

2

$f(x)=\frac{1}{x^2-2}$ $(x\ne \pm \sqrt{2})$

$f(5)=\frac{1}{5^2-2}=\frac{1}{25-2}=\frac{1}{23}$

3

$f(x)=\sqrt[3]{x}$ is defined for all real numbers.

$f(27)=\sqrt[3]{27}=3$ because $3^3=27.$

4

$a)f(1)=1$

$b)f(-3)=3$

$c)f(-\frac{4}{3})=\frac{4}{3}$

5

$a)f\left(\frac{1}{2}\right)=1/2+|1/2|=1$

$b)f(2)=2+|2|=4$

$c)f(-4)=-4+|-4|=-4+4=0$

$d)f(-5)=-5+|-5|=-5+5=0$

6

$f(x)=2x+x^2-5$

$a)f(1)=2(1)+(1{)}^2-5=-2$

$b)f(-1)=2(-1)+(-1{)}^2-5=-6$

7

$f(x)=\sqrt[4]{x}$

$x\ge 0$

$f(16)=2,$ because $2^4=16$

8

$f(x)=-f(x)$ is said to be an even function for all numbers x.

$f(x)=-f(-x)$ is said to be an odd function for all x.

$a)f(x)=x$ is odd.

$b)f(x)=x^2$ is even.

$c)f(x)=x^3$ is odd.

$d)f(x)=\frac{1}{x}$ if $x\ne 0$ and $f(0)=0$ is odd.

9

Let $f_e(x)=\frac{f(x)+f(-x)}{2}$ and $f_o(x)=\frac{f(x)-f(-x)}{2}$ be the even and odd functions respectively.

Then $f_e(x)+f_o(x)=\frac{f(x)+f(-x)+f(x)-f(-x)}{2}=f_e(x)$

10

a) Odd

b) Even

c) Odd

d) Odd

e) Even

f) Even

g) Even

11

$a)$ $f_a(x)=\frac{f(x)-f(-x)}{2}$ $f_b(x)=\frac{f(x)-f(-x)}{2}$

$f_a(x)+f_b(x)=\frac{f(x)-f(-x)+f(x)-f(-x)}{2}$

It follows that $f_a(x)+f_b(x)=f(x)-f(-x)$ (The odd function)

$b)$ $f_a(x)=\frac{f(x)+f(-x)}{2}$ $f_b(x)=\frac{f(x)+f(-x)}{2}$

$f_a(x)+f_b(x)=\frac{f(x)+f(-x)+f(x)+f(-x)}{2}$

It follows that $f_a(x)+f_b(x)=f(x)+f(-x)$ (The even function)