QuestionFind the inverse of the one-to-one function $f(x)=\frac{2}{5 x+3}$ .

Studdy Solution

STEP 1

Assumptions1. The function f is one-to-one, which means that for every y in the range, there is exactly one x in the domain such that f(x) = y. . We are asked to find the inverse of the function f.

STEP 2

The first step in finding the inverse of a function is to replace the function notation f(x) with y. This gives us a new equation $y = \frac{2}{5x +}$

STEP 3

The next step is to switch the roles of y and x. This means we replace every x in the equation with y and every y with x. This gives us $x = \frac{2}{5y +3}$

STEP 4

Now we need to solve this equation for y, which will give us the inverse function. First, we can multiply both sides of the equation by (y +3) to get rid of the fraction $(y +3)x =2$

STEP 5

Next, distribute the x on the left side of the equation $5xy +3x =2$

STEP 6

Now, isolate the term with y on one side of the equation $5xy =2 -3x$

STEP 7

Finally, divide both sides of the equation by5x to solve for y $y = \frac{2 -3x}{5x}$ This is the inverse of the function f.

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QuestionFind the inverse of the one-to-one function $f(x)=\frac{2}{5 x+3}$ .

Studdy Solution

STEP 1

Assumptions1. The function f is one-to-one, which means that for every y in the range, there is exactly one x in the domain such that f(x) = y. . We are asked to find the inverse of the function f.

STEP 2

The first step in finding the inverse of a function is to replace the function notation f(x) with y. This gives us a new equation $y = \frac{2}{5x +}$

STEP 3

The next step is to switch the roles of y and x. This means we replace every x in the equation with y and every y with x. This gives us $x = \frac{2}{5y +3}$

STEP 4

Now we need to solve this equation for y, which will give us the inverse function. First, we can multiply both sides of the equation by (y +3) to get rid of the fraction $(y +3)x =2$

STEP 5

Next, distribute the x on the left side of the equation $5xy +3x =2$

STEP 6

Now, isolate the term with y on one side of the equation $5xy =2 -3x$

STEP 7

Finally, divide both sides of the equation by5x to solve for y $y = \frac{2 -3x}{5x}$ This is the inverse of the function f.

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QuestionFind the inverse of the one-to-one function f(x)=25x+3f(x)=\frac{2}{5 x+3}f(x)=5x+32​.

Studdy Solution

STEP 1

Assumptions1. The function f is one-to-one, which means that for every y in the range, there is exactly one x in the domain such that f(x) = y. . We are asked to find the inverse of the function f.

STEP 2

The first step in finding the inverse of a function is to replace the function notation f(x) with y. This gives us a new equationy=25x+y = \frac{2}{5x +}y=5x+2​

STEP 3

The next step is to switch the roles of y and x. This means we replace every x in the equation with y and every y with x. This gives usx=25y+3x = \frac{2}{5y +3}x=5y+32​

STEP 4

Now we need to solve this equation for y, which will give us the inverse function. First, we can multiply both sides of the equation by (y +3) to get rid of the fraction(y+3)x=2(y +3)x =2(y+3)x=2

STEP 5

Next, distribute the x on the left side of the equation5xy+3x=25xy +3x =25xy+3x=2

STEP 6

Now, isolate the term with y on one side of the equation5xy=2−3x5xy =2 -3x5xy=2−3x

STEP 7

Finally, divide both sides of the equation by5x to solve for yy=2−3x5xy = \frac{2 -3x}{5x}y=5x2−3x​This is the inverse of the function f.

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QuestionFind the inverse of the one-to-one function $f(x)=\frac{2}{5 x+3}$ .

The first step in finding the inverse of a function is to replace the function notation f(x) with y. This gives us a new equation $y = \frac{2}{5x +}$

The next step is to switch the roles of y and x. This means we replace every x in the equation with y and every y with x. This gives us $x = \frac{2}{5y +3}$

Now we need to solve this equation for y, which will give us the inverse function. First, we can multiply both sides of the equation by (y +3) to get rid of the fraction $(y +3)x =2$

Next, distribute the x on the left side of the equation $5xy +3x =2$

Now, isolate the term with y on one side of the equation $5xy =2 -3x$

Finally, divide both sides of the equation by5x to solve for y $y = \frac{2 -3x}{5x}$ This is the inverse of the function f.