Math / Chem

Math

Chem

Chem 7.31 - 7.45

HW2.7 Q1

$f(x)=6x$

$g(x)=\frac{x}{6}$

$f(g(x))=f(\frac{x}{6})\\=6(\frac{x}{6})\\=6x/6\\=x$

$g(f(x))=g(6x)\\=\frac{6x}{6}\\=x$

HW2.7 Q2

$f(x)=5x-6$

$g(x)=\frac{x+5}{6}$

$f(g(x)) = f(\frac{x+5}{6})\\=5(\frac{x+5}{6})-6\\=\frac{5x+25}{6}-6\\=\frac{5x+25}{6}-\frac{6\times6}{6}\\=\frac{5x+25}{6}-\frac{36}{6}\\=\color{blue}{\frac{5x-11}{6}}$

$g(f(x)) = g(5x-6)\\=\frac{5x-6+5}{6}\\=\color{blue}{\frac{5x-1}{6}}$

HW2.7 Q4

$f(x)=3x$

To find $f^{-1}(x)$ :

1) Replace $f(x)$ with $y$ .
2) Interachange $x$ and $y$ .
3) Solve for $y$ .
4) Replace $y$ with $f^{-1}(x)$ .

1) $y=3x$
2) $x=3y$
3) $y=\frac{x}{3}$
4) $f^{-1}(x)=\frac{x}{3}$

HW2.7 Q5

$f(x)=7x+2$

$y=7x+2$
$x=7y+2$
$7y=x-2$
$y=\frac{x-2}{7}$
$\color{blue}{f^{-1}(x)=\frac{x-2}{7}}$

HW2.7 Q6

$f(x)=x^3-2$

$y=x^3-2$
$x=y^3-2$
$y^3=x+2$

$y=\sqrt[3]{x+2}$

$\color{blue}{f^{-1}(x)=\sqrt[3]{x+2}}$

HW2.7 Q7

$f(x)=\frac{19}{x}-1$

$y=\frac{19}{x}-1$

$x=\frac{19}{y}-1$

$\frac{19}{y}=x+1$

$y=\frac{19}{x+1}$

$\color{blue}{f^{-1}(x)=\frac{19}{x+1}}$

HW2.7 Q8

$f(x)=\frac{5x+6}{x-1}$

$y=\frac{5x+6}{x-1}$

$x=\frac{5y+6}{y-1}$

$x(y-1)=5y+6$

$xy-x=5y+6$

$5y = xy-x-6$

$5y -xy = -x-6$

$y(5-x) = -x-6$

$y = \frac{-x-6}{5-x}$

$\color{blue}{f^{-1}(x)=\frac{-x-6}{5-x}}$