Question

1. $f(x)=-\frac{5}{3}x - \frac{10}{3}$
2. $g(x)=-x + 3$

$\frac{x - 3}{-1}=\frac{-1y}{-1}$
$-x + 3 = y$
$g^{-1}(x)=-x + 3$

1. $f(x)=5 - \frac{7}{4}x$

$x - 5 = -\frac{7}{4}x$

1. $f(x)=-4 + \frac{1}{5}x$

$x + 4 = \frac{1}{5}y$
$x + 25 = y$
$f^{-1}(x)=5x + 25$

Explanation:

Step1: Replace $f(x)$ with $y$

$y = -\frac{5}{3}x - \frac{10}{3}$

Step2: Swap $x$ and $y$

$x = -\frac{5}{3}y - \frac{10}{3}$

Step3: Isolate the term with $y$

$x + \frac{10}{3} = -\frac{5}{3}y$

Step4: Solve for $y$

Multiply both sides by $-\frac{3}{5}$:
$y = -\frac{3}{5}x - 2$
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Step1: (Verify/redo $g(x)$) Replace $g(x)$ with $y$

$y = -x + 3$

Step2: Swap $x$ and $y$

$x = -y + 3$

Step3: Solve for $y$

$y = -x + 3$
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Step1: Replace $f(x)$ with $y$

$y = 5 - \frac{7}{4}x$

Step2: Swap $x$ and $y$

$x = 5 - \frac{7}{4}y$

Step3: Isolate the term with $y$

$\frac{7}{4}y = 5 - x$

Step4: Solve for $y$

Multiply both sides by $\frac{4}{7}$:
$y = \frac{20}{7} - \frac{4}{7}x$
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Step1: (Verify/redo $f(x)$) Replace $f(x)$ with $y$

$y = -4 + \frac{1}{5}x$

Step2: Swap $x$ and $y$

$x = -4 + \frac{1}{5}y$

Step3: Isolate the term with $y$

$x + 4 = \frac{1}{5}y$

Step4: Solve for $y$

Multiply both sides by 5:
$y = 5x + 20$

Answer:

1. $f^{-1}(x) = -\frac{3}{5}x - 2$
2. $g^{-1}(x) = -x + 3$
3. $f^{-1}(x) = -\frac{4}{7}x + \frac{20}{7}$
4. $f^{-1}(x) = 5x + 20$