Math / Algebra

QuestionDirections: Find the inverse of the following functions. Be sure to show all work and use proper notation.
8. $f(x)=\frac{x-3}{x+2}$
9. $h(x)=\frac{4 x-1}{2 x+3}$ $\begin{array}{l} x=\frac{y-3}{y+2} \\ x(y+2)=y-3 \\ x y+2 x=y-3 \\ 2 x-3=y-x y \\ 2 x-3=y(1-x) \\ y=\frac{2 x-3}{1-x} \end{array}$ $x=\frac{4 y-1}{2 y+3}$ $\begin{array}{l} x(2 y+3)=4 y-1 \\ 2 x y+3 x=4 y-1 \\ 3 x+1=4 y-2 x y \\ 3 x+1=y(4-2 x) \end{array}$ $h(x)^{-1}=\sqrt{y=\frac{3 x+1}{4-2 x}}$
10. $y=\frac{x+4}{x-5}$
11. $g(x)=\frac{2 x+1}{3 x+7}$

Studdy Solution

STEP 1

1. We are tasked with finding the inverse of given rational functions.
2. We need to show all work and use proper notation for inverse functions.

STEP 2

1. Find the inverse of $f(x) = \frac{x-3}{x+2}$ .
2. Find the inverse of $h(x) = \frac{4x-1}{2x+3}$ .
3. Find the inverse of $y = \frac{x+4}{x-5}$ .
4. Find the inverse of $g(x) = \frac{2x+1}{3x+7}$ .

STEP 3

To find the inverse of $f(x) = \frac{x-3}{x+2}$ , start by setting $y = \frac{x-3}{x+2}$ .

STEP 4

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{y-3}{y+2}$

STEP 5

Multiply both sides by $y+2$ to eliminate the fraction:
$x(y+2) = y-3$

STEP 6

Distribute $x$ on the left side:
$xy + 2x = y - 3$

STEP 7

Rearrange the terms to isolate $y$ :
$xy - y = -2x - 3$
Factor out $y$ on the left side:
$y(x-1) = -2x - 3$

STEP 8

Solve for $y$ :
$y = \frac{-2x - 3}{x - 1}$
Thus, the inverse function is:
$f^{-1}(x) = \frac{-2x - 3}{x - 1}$

STEP 9

To find the inverse of $h(x) = \frac{4x-1}{2x+3}$ , start by setting $y = \frac{4x-1}{2x+3}$ .

STEP 10

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{4y-1}{2y+3}$

STEP 11

Multiply both sides by $2y+3$ to eliminate the fraction:
$x(2y+3) = 4y - 1$

STEP 12

Distribute $x$ on the left side:
$2xy + 3x = 4y - 1$

STEP 13

Rearrange the terms to isolate $y$ :
$2xy - 4y = -3x - 1$
Factor out $y$ on the left side:
$y(2x - 4) = -3x - 1$

STEP 14

Solve for $y$ :
$y = \frac{-3x - 1}{2x - 4}$
Thus, the inverse function is:
$h^{-1}(x) = \frac{-3x - 1}{2x - 4}$

STEP 15

To find the inverse of $y = \frac{x+4}{x-5}$ , start by setting $y = \frac{x+4}{x-5}$ .

STEP 16

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{y+4}{y-5}$

STEP 17

Multiply both sides by $y-5$ to eliminate the fraction:
$x(y-5) = y + 4$

STEP 18

Distribute $x$ on the left side:
$xy - 5x = y + 4$

STEP 19

Rearrange the terms to isolate $y$ :
$xy - y = 5x + 4$
Factor out $y$ on the left side:
$y(x - 1) = 5x + 4$

STEP 20

Solve for $y$ :
$y = \frac{5x + 4}{x - 1}$
Thus, the inverse function is:
$y^{-1}(x) = \frac{5x + 4}{x - 1}$

STEP 21

To find the inverse of $g(x) = \frac{2x+1}{3x+7}$ , start by setting $y = \frac{2x+1}{3x+7}$ .

STEP 22

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{2y+1}{3y+7}$

STEP 23

Multiply both sides by $3y+7$ to eliminate the fraction:
$x(3y+7) = 2y + 1$

STEP 24

Distribute $x$ on the left side:
$3xy + 7x = 2y + 1$

STEP 25

Rearrange the terms to isolate $y$ :
$3xy - 2y = 1 - 7x$
Factor out $y$ on the left side:
$y(3x - 2) = 1 - 7x$

STEP 26

Solve for $y$ :
$y = \frac{1 - 7x}{3x - 2}$
Thus, the inverse function is:
$g^{-1}(x) = \frac{1 - 7x}{3x - 2}$
The inverses of the functions are:
1. $f^{-1}(x) = \frac{-2x - 3}{x - 1}$
2. $h^{-1}(x) = \frac{-3x - 1}{2x - 4}$
3. $y^{-1}(x) = \frac{5x + 4}{x - 1}$
4. $g^{-1}(x) = \frac{1 - 7x}{3x - 2}$

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Studdy Solution

STEP 1

1. We are tasked with finding the inverse of given rational functions.2. We need to show all work and use proper notation for inverse functions.

STEP 2

STEP 3

To find the inverse of f(x)=x−3x+2 f(x) = \frac{x-3}{x+2} f(x)=x+2x−3​, start by setting y=x−3x+2 y = \frac{x-3}{x+2} y=x+2x−3​.

STEP 4

Swap x x x and y y y to solve for y y y:x=y−3y+2 x = \frac{y-3}{y+2} x=y+2y−3​

STEP 5

Multiply both sides by y+2 y+2 y+2 to eliminate the fraction:x(y+2)=y−3 x(y+2) = y-3 x(y+2)=y−3

STEP 6

Distribute x x x on the left side:xy+2x=y−3 xy + 2x = y - 3 xy+2x=y−3

STEP 7

Rearrange the terms to isolate y y y:xy−y=−2x−3 xy - y = -2x - 3 xy−y=−2x−3Factor out y y y on the left side:y(x−1)=−2x−3 y(x-1) = -2x - 3 y(x−1)=−2x−3

STEP 8

Solve for y y y:y=−2x−3x−1 y = \frac{-2x - 3}{x - 1} y=x−1−2x−3​Thus, the inverse function is:f−1(x)=−2x−3x−1 f^{-1}(x) = \frac{-2x - 3}{x - 1} f−1(x)=x−1−2x−3​

STEP 9

To find the inverse of h(x)=4x−12x+3 h(x) = \frac{4x-1}{2x+3} h(x)=2x+34x−1​, start by setting y=4x−12x+3 y = \frac{4x-1}{2x+3} y=2x+34x−1​.

STEP 10

Swap x x x and y y y to solve for y y y:x=4y−12y+3 x = \frac{4y-1}{2y+3} x=2y+34y−1​

STEP 11

Multiply both sides by 2y+3 2y+3 2y+3 to eliminate the fraction:x(2y+3)=4y−1 x(2y+3) = 4y - 1 x(2y+3)=4y−1

STEP 12

Distribute x x x on the left side:2xy+3x=4y−1 2xy + 3x = 4y - 1 2xy+3x=4y−1

STEP 13

Rearrange the terms to isolate y y y:2xy−4y=−3x−1 2xy - 4y = -3x - 1 2xy−4y=−3x−1Factor out y y y on the left side:y(2x−4)=−3x−1 y(2x - 4) = -3x - 1 y(2x−4)=−3x−1

STEP 14

Solve for y y y:y=−3x−12x−4 y = \frac{-3x - 1}{2x - 4} y=2x−4−3x−1​Thus, the inverse function is:h−1(x)=−3x−12x−4 h^{-1}(x) = \frac{-3x - 1}{2x - 4} h−1(x)=2x−4−3x−1​

STEP 15

To find the inverse of y=x+4x−5 y = \frac{x+4}{x-5} y=x−5x+4​, start by setting y=x+4x−5 y = \frac{x+4}{x-5} y=x−5x+4​.

STEP 16

Swap x x x and y y y to solve for y y y:x=y+4y−5 x = \frac{y+4}{y-5} x=y−5y+4​

STEP 17

Multiply both sides by y−5 y-5 y−5 to eliminate the fraction:x(y−5)=y+4 x(y-5) = y + 4 x(y−5)=y+4

STEP 18

Distribute x x x on the left side:xy−5x=y+4 xy - 5x = y + 4 xy−5x=y+4

STEP 19

Rearrange the terms to isolate y y y:xy−y=5x+4 xy - y = 5x + 4 xy−y=5x+4Factor out y y y on the left side:y(x−1)=5x+4 y(x - 1) = 5x + 4 y(x−1)=5x+4

STEP 20

Solve for y y y:y=5x+4x−1 y = \frac{5x + 4}{x - 1} y=x−15x+4​Thus, the inverse function is:y−1(x)=5x+4x−1 y^{-1}(x) = \frac{5x + 4}{x - 1} y−1(x)=x−15x+4​

STEP 21

To find the inverse of g(x)=2x+13x+7 g(x) = \frac{2x+1}{3x+7} g(x)=3x+72x+1​, start by setting y=2x+13x+7 y = \frac{2x+1}{3x+7} y=3x+72x+1​.

STEP 22

Swap x x x and y y y to solve for y y y:x=2y+13y+7 x = \frac{2y+1}{3y+7} x=3y+72y+1​

STEP 23

Multiply both sides by 3y+7 3y+7 3y+7 to eliminate the fraction:x(3y+7)=2y+1 x(3y+7) = 2y + 1 x(3y+7)=2y+1

STEP 24

Distribute x x x on the left side:3xy+7x=2y+1 3xy + 7x = 2y + 1 3xy+7x=2y+1

STEP 25

Rearrange the terms to isolate y y y:3xy−2y=1−7x 3xy - 2y = 1 - 7x 3xy−2y=1−7xFactor out y y y on the left side:y(3x−2)=1−7x y(3x - 2) = 1 - 7x y(3x−2)=1−7x

STEP 26

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1. We are tasked with finding the inverse of given rational functions.
2. We need to show all work and use proper notation for inverse functions.

To find the inverse of $f(x) = \frac{x-3}{x+2}$ , start by setting $y = \frac{x-3}{x+2}$ .

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{y-3}{y+2}$

Multiply both sides by $y+2$ to eliminate the fraction:
$x(y+2) = y-3$

Distribute $x$ on the left side:
$xy + 2x = y - 3$

Rearrange the terms to isolate $y$ :
$xy - y = -2x - 3$
Factor out $y$ on the left side:
$y(x-1) = -2x - 3$

Solve for $y$ :
$y = \frac{-2x - 3}{x - 1}$
Thus, the inverse function is:
$f^{-1}(x) = \frac{-2x - 3}{x - 1}$

To find the inverse of $h(x) = \frac{4x-1}{2x+3}$ , start by setting $y = \frac{4x-1}{2x+3}$ .

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{4y-1}{2y+3}$

Multiply both sides by $2y+3$ to eliminate the fraction:
$x(2y+3) = 4y - 1$

Distribute $x$ on the left side:
$2xy + 3x = 4y - 1$

Rearrange the terms to isolate $y$ :
$2xy - 4y = -3x - 1$
Factor out $y$ on the left side:
$y(2x - 4) = -3x - 1$

Solve for $y$ :
$y = \frac{-3x - 1}{2x - 4}$
Thus, the inverse function is:
$h^{-1}(x) = \frac{-3x - 1}{2x - 4}$

To find the inverse of $y = \frac{x+4}{x-5}$ , start by setting $y = \frac{x+4}{x-5}$ .

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{y+4}{y-5}$

Multiply both sides by $y-5$ to eliminate the fraction:
$x(y-5) = y + 4$

Distribute $x$ on the left side:
$xy - 5x = y + 4$

Rearrange the terms to isolate $y$ :
$xy - y = 5x + 4$
Factor out $y$ on the left side:
$y(x - 1) = 5x + 4$

Solve for $y$ :
$y = \frac{5x + 4}{x - 1}$
Thus, the inverse function is:
$y^{-1}(x) = \frac{5x + 4}{x - 1}$

To find the inverse of $g(x) = \frac{2x+1}{3x+7}$ , start by setting $y = \frac{2x+1}{3x+7}$ .

Swap $x$ and $y$ to solve for $y$ :
$x = \frac{2y+1}{3y+7}$

Multiply both sides by $3y+7$ to eliminate the fraction:
$x(3y+7) = 2y + 1$

Distribute $x$ on the left side:
$3xy + 7x = 2y + 1$

Rearrange the terms to isolate $y$ :
$3xy - 2y = 1 - 7x$
Factor out $y$ on the left side:
$y(3x - 2) = 1 - 7x$