inverse function

Question

Find the inverse of the function.

1)f(x)=-x+7
2)f(x)=3x+4
3)f(x)=2x-4
4)f(x)=x³-5
5)f(x)=x²-2

Answer 1

1) f(x)=-x+7

Given f(x)=-x+7

To finding a formula for ƒ⁻¹, if it exists, is to solve the equation y = ƒ(x) for x.

f(x)=-x+7

we must solve the equation y =   -x+7 for x.

y =-x+7                                        (Subtract each side with 7)

y-7 = -x +7 -7                              (Simplify)

y -7 = -x                                       (Multiply each side by (   -   ) )

-( y -7 ) = - (-x)                            (Simplify)

-y+7 = x    

x = -y +7

Thus the inverse ƒ⁻¹ (y) = -y +7

2)f(x)=3x+4

Exchange the sides of the equation and solve for x

3x+4 = y

Subtract 4 to each side

3x+4-4 = y-4

3x = y-4

Divide   each side by 3

3x/3 = (y-4)/3

x = (y-4)/3

Exchange the variables

y = (x-4)/3

That function is inverse of y = 3x+4

3) f(x)=2x-4

One approach to finding a formula for ƒ⁻¹, if it exists, is to solve the equation y = ƒ(x) for x.

f(x)=2x-4

then we must solve the equation y = 2x-4   for x:

y=2x-4

Add 4 to each side

y+4=2x-4+4

y+4=2x

Divide each side by 2

(y+4)/2=2x/2

x=(y+4)/2

Exchange the variables

y=2x-4

That function is inverse of y =2x-4

4) f(x)=x³-5

Given f(x)=x³-5

To finding a formula for ƒ⁻¹, if it exists, is to solve the equation y = ƒ(x) for x.

y =  x³-5

we must solve the equationy =  x³-5 for x.

y =  x³-5                                   (Add 5 to each side)

y+5 =x³-5+5                            (Simplify)  

y+5 =x³

³√(y+5)=x

 x= ³√(y+5)

Thus the inverse ƒ⁻¹ (y) = ³√(y+5)

5)f(x)=x²-2

Given f(x)=x²-2

To finding a formula for ƒ⁻¹, if it exists, is to solve the equation y = ƒ(x) for x.

f(x)^-1=x²-2

we must solve the equation y =   x²-2 for x.

y+2 =x²-2+2                            (Add each side by 2)

y + 2 = x²                                 (Simplify)

x² = y + 2                                                

sqareing on each side                        

√x² = √(y + 2)  

x = √(y + 2)

Thus the inverse ƒ⁻¹ (y) = √(y + 2)

Comments

1). The inverse function is ƒ⁻¹ (x) = - x + 7.

2). The inverse function is ƒ⁻¹ (x) = (x - 4) / 3.

3). The inverse function is ƒ⁻¹ (x) = (x + 4)/2

4). The inverse function is ƒ⁻¹ (x) = ³√(x + 5)

5). The inverse function is ƒ⁻¹ (x) = √(x + 2).