Find the inverse of the function.

Question

 

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

3.

Given function is f(x) = 2x - 4

Replace f(x) with y

y = 2x - 4

Then switch x for y and y for x

y = 2x - 4

x = 2y - 4

Solve for y:-

x = 2y - 4

Add 4 each side

x+4 = 2y - 4+4

x+4 = 2y

Divide each side by 2

(x+4)/2 = 2y

x/2+2 = y

Write in function notation, ƒ⁻¹. to represent the inverse function ƒ

ƒ⁻¹(x)= y

Substitute x/2+2 = y in ƒ⁻¹(x)= y

ƒ⁻¹(x) = x/2+2

The inverse of functon ƒ(x) = 2x - 4 is ƒ⁻¹(x) = x/2+2

Comments

Function is f(x ) = 2x - 4

y = 2x - 4

To find the inverse function first solve for x and then switch x for y and y for x.

y + 4 = 2x

(y + 4) / 2 = x

x = (y + 4) / 2

Switch x for y and y for x.

y = (x + 4) / 2

ƒ⁻¹(x ) = (x + 4) / 2

Answer 2

2)

The given equation is f(x) = 3x + 4

Let f(x) = y

f(x)= 3x + 4 = y

Then f(x) = y and 3x + 4 = y

f(x) = y

x = ƒ⁻¹(y)-----------------(1)

3x + 4 = y

Subtract 4 from each side

3x + 4 - 4 = y - 4

3x = y - 4

Divide each side by 3

x = y - 4 /3

The value of x = y - 4 /3 is substitute in equation (1)

y - 4 /3 = ƒ⁻¹(y)

ƒ⁻¹(y) = y - 4 /3

Replace y with x, then

ƒ⁻¹(x) = x - 4 /3

The inverse of the function f(x)=3x+4 is ƒ⁻¹(x) = x - 4 /3

Comments

The inverse of the function f(x)=3x+4 is ƒ⁻¹(x) = (x - 4)/3.

Answer 3

1) f(x) = -x+7

Given that f(x) = -x+7

Let f(x) = y

 y = -x+7

from  f(x) = y

x = ƒ⁻¹(y) --------------(1)

y = -x+7

subtract 7 from each side

y-7 = -x+7-7

y-7 = -x

x = 7-y -------------(2)

From (1) and (2)

ƒ⁻¹(y) = 7-y

Replace y with x , then

ƒ⁻¹(x) = 7-x

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

Comments

f(x ) = -x + 7

y = -x + 7

Add x to each side.

x + y = x - x + 7

x + y = 0 + 7

x + y = 7

Subtract y from each side.

x + y - y = 7 - y

x + 0 = 7 - y

x = 7 - y

Replace y with x , then y = 7 - x.

Therefore f^-1(x ) = 7 - x.

Answer 4

5) f(x) = x²  - 2

Replace f(x) by y

y = x²-2

Add 2 to each side

y + 2 = x² 

± sqrt (y+2) = x

Now switch x and y

± sqrt (x+2) = y

Replace y by ƒ⁻¹(x)

ƒ⁻¹(x) = ± sqrt (x+2)

The inverse of  f(x) = x²  - 2 is ƒ⁻¹(x)=± sqrt (x+2)

Answer 5

4)f(x)=x^3-5

Replace f(x) by y

f(x)=y

Then x=ƒ⁻¹(y)--------(1)

y=x^3-5

Add 5 to each side

y+5=x^3

(y+5)^1/3=x-------(2)

From (1) and (2)

ƒ⁻¹(y)=(y+5)^1/3

Replace y by x

ƒ⁻¹(x)=(x+5)^1/3

 

Answer 6

3.

Given function is f(x) = 2x - 4

Replace f(x) with y

y = 2x - 4

2x= y +4

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

x=(y +4) /2

x =y/2 +2

ƒ⁻¹(x)= x/2 +2

Write in function notation, ƒ⁻¹. to represent the inverse function ƒ

ƒ⁻¹(x)= y

Substitute x/2+2 = y in ƒ⁻¹(x)= y

ƒ⁻¹(x) = x/2+2

The inverse of functon ƒ(x) = 2x - 4 is ƒ⁻¹(x) = x/2+2

 

Comments

The inverse of functon ƒ(x) = 2x - 4 is ƒ⁻¹(x) = (x + 4)/2.