Respuesta :

calculista

Answer:

Part 1)  [tex]f^{-1}(x)=3(x+17)/2[/tex] ----> [tex]f(x)=\frac{2x}{3}-17[/tex]

Part 2) [tex]f^{-1}(x)=x+10[/tex] -----> [tex]f(x)=x-10[/tex]

Part 3) [tex]f^{-1}(x)=\frac{x^{3}}{2}[/tex] ----> [tex]f(x)=\sqrt[3]{2x}[/tex]

Part 4) [tex]f^{-1}(x)=5x[/tex] ----> [tex]f(x)=x/5[/tex]

Step-by-step explanation:

Part 1) we have

[tex]f(x)=\frac{2x}{3}-17[/tex]

Find the inverse

Let

y=f(x)

[tex]y=\frac{2x}{3}-17[/tex]

Exchange the variables, x for y and y for x

[tex]x=\frac{2y}{3}-17[/tex]

Isolate the variable y

Adds 17 both sides

[tex]x+17=\frac{2y}{3}[/tex]

Multiply by 3 both sides

[tex]3(x+17)=2y[/tex]

Divide by 2 both sides

[tex]y=3(x+17)/2[/tex]

Let

[tex]f^{-1}(x)=y[/tex]

so

[tex]f^{-1}(x)=3(x+17)/2[/tex]

Part 2) we have

[tex]f(x)=x-10[/tex]

Find the inverse

Let

y=f(x)

[tex]y=x-10[/tex]

Exchange the variables, x for y and y for x

[tex]x=y-10[/tex]

Isolate the variable y

Adds 10 both sides

[tex]y=x+10[/tex]

Let

[tex]f^{-1}(x)=y[/tex]

so

[tex]f^{-1}(x)=x+10[/tex]

Part 3) we have

[tex]f(x)=\sqrt[3]{2x}[/tex]

Find the inverse

Let

y=f(x)

[tex]y=\sqrt[3]{2x}[/tex]

Exchange the variables, x for y and y for x

[tex]x=\sqrt[3]{2y}[/tex]

Isolate the variable y

elevated to the cube both sides

[tex]x^{3}=2y[/tex]

Divide by 2 both sides

[tex]y=\frac{x^{3}}{2}[/tex]

Let

[tex]f^{-1}(x)=y[/tex]

so

[tex]f^{-1}(x)=\frac{x^{3}}{2}[/tex]

Part 4) we have

[tex]f(x)=x/5[/tex]

Find the inverse

Let

y=f(x)

[tex]y=x/5[/tex]

Exchange the variables, x for y and y for x

[tex]x=y/5[/tex]

Isolate the variable y

Multiply by 5 both sides

[tex]y=5x[/tex]

Let

[tex]f^{-1}(x)=y[/tex]

so

[tex]f^{-1}(x)=5x[/tex]

The inverse of a function is shown in the picture we can calculate by interchanging the value of f(x) and x.

What is a function?

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

We have:

Functions are shown in the picture.

We have to find the inverse of a function.

[tex]\rm f(x) = \dfrac{2x}{3}-17[/tex]

To find the inverse of a function plug in the place of f(x)→x and x→f(x) ⁻¹

[tex]\rm x = \dfrac{2f^-^1(x)}{3}-17[/tex]

f(x) ⁻¹ = 3(x + 17)/2

Similarly, we can find the rest of the inverse of the function as follows:

f(x) = x - 10

f(x) ⁻¹ = x + 10

f(x) = ∛2x

f(x) ⁻¹ = x³/2

f(x) = x/5

f(x) ⁻¹ = 5x

Thus, the inverse of a function is shown in the picture we can calculate by interchanging the value of f(x) and x.

Learn more about the function here:

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