Precalculus Examples
f(x)=6−4x
Step 1
Write f(x)=6−4x as an equation.
y=6−4x
Step 2
Interchange the variables.
x=6−4y
Step 3
Step 3.1
Rewrite the equation as 6−4y=x.
6−4y=x
Step 3.2
Subtract 6 from both sides of the equation.
−4y=x−6
Step 3.3
Divide each term in −4y=x−6 by −4 and simplify.
Step 3.3.1
Divide each term in −4y=x−6 by −4.
−4y−4=x−4+−6−4
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Cancel the common factor of −4.
Step 3.3.2.1.1
Cancel the common factor.
−4y−4=x−4+−6−4
Step 3.3.2.1.2
Divide y by 1.
y=x−4+−6−4
y=x−4+−6−4
y=x−4+−6−4
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Simplify each term.
Step 3.3.3.1.1
Move the negative in front of the fraction.
y=−x4+−6−4
Step 3.3.3.1.2
Cancel the common factor of −6 and −4.
Step 3.3.3.1.2.1
Factor −2 out of −6.
y=−x4+−2(3)−4
Step 3.3.3.1.2.2
Cancel the common factors.
Step 3.3.3.1.2.2.1
Factor −2 out of −4.
y=−x4+−2⋅3−2⋅2
Step 3.3.3.1.2.2.2
Cancel the common factor.
y=−x4+−2⋅3−2⋅2
Step 3.3.3.1.2.2.3
Rewrite the expression.
y=−x4+32
y=−x4+32
y=−x4+32
y=−x4+32
y=−x4+32
y=−x4+32
y=−x4+32
Step 4
Replace y with f−1(x) to show the final answer.
f−1(x)=−x4+32
Step 5
Step 5.1
To verify the inverse, check if f−1(f(x))=x and f(f−1(x))=x.
Step 5.2
Evaluate f−1(f(x)).
Step 5.2.1
Set up the composite result function.
f−1(f(x))
Step 5.2.2
Evaluate f−1(6−4x) by substituting in the value of f into f−1.
f−1(6−4x)=−6−4x4+32
Step 5.2.3
Simplify terms.
Step 5.2.3.1
Cancel the common factor of 6−4x and 4.
Step 5.2.3.1.1
Factor 2 out of 6.
f−1(6−4x)=−2(3)−4x4+32
Step 5.2.3.1.2
Factor 2 out of −4x.
f−1(6−4x)=−2(3)+2(−2x)4+32
Step 5.2.3.1.3
Factor 2 out of 2(3)+2(−2x).
f−1(6−4x)=−2(3−2x)4+32
Step 5.2.3.1.4
Cancel the common factors.
Step 5.2.3.1.4.1
Factor 2 out of 4.
f−1(6−4x)=−2(3−2x)2⋅2+32
Step 5.2.3.1.4.2
Cancel the common factor.
f−1(6−4x)=−2(3−2x)2⋅2+32
Step 5.2.3.1.4.3
Rewrite the expression.
f−1(6−4x)=−3−2x2+32
f−1(6−4x)=−3−2x2+32
f−1(6−4x)=−3−2x2+32
Step 5.2.3.2
Combine the numerators over the common denominator.
f−1(6−4x)=−(3−2x)+32
f−1(6−4x)=−(3−2x)+32
Step 5.2.4
Simplify each term.
Step 5.2.4.1
Apply the distributive property.
f−1(6−4x)=−1⋅3−(−2x)+32
Step 5.2.4.2
Multiply −1 by 3.
f−1(6−4x)=−3−(−2x)+32
Step 5.2.4.3
Multiply −2 by −1.
f−1(6−4x)=−3+2x+32
f−1(6−4x)=−3+2x+32
Step 5.2.5
Simplify terms.
Step 5.2.5.1
Combine the opposite terms in −3+2x+3.
Step 5.2.5.1.1
Add −3 and 3.
f−1(6−4x)=2x+02
Step 5.2.5.1.2
Add 2x and 0.
f−1(6−4x)=2x2
f−1(6−4x)=2x2
Step 5.2.5.2
Cancel the common factor of 2.
Step 5.2.5.2.1
Cancel the common factor.
f−1(6−4x)=2x2
Step 5.2.5.2.2
Divide x by 1.
f−1(6−4x)=x
f−1(6−4x)=x
f−1(6−4x)=x
f−1(6−4x)=x
Step 5.3
Evaluate f(f−1(x)).
Step 5.3.1
Set up the composite result function.
f(f−1(x))
Step 5.3.2
Evaluate f(−x4+32) by substituting in the value of f−1 into f.
f(−x4+32)=6−4(−x4+32)
Step 5.3.3
Simplify each term.
Step 5.3.3.1
Apply the distributive property.
f(−x4+32)=6−4(−x4)−4(32)
Step 5.3.3.2
Cancel the common factor of 4.
Step 5.3.3.2.1
Move the leading negative in −x4 into the numerator.
f(−x4+32)=6−4−x4−4(32)
Step 5.3.3.2.2
Factor 4 out of −4.
f(−x4+32)=6+4(−1)(−x4)−4(32)
Step 5.3.3.2.3
Cancel the common factor.
f(−x4+32)=6+4⋅(−1−x4)−4(32)
Step 5.3.3.2.4
Rewrite the expression.
f(−x4+32)=6−1(−x)−4(32)
f(−x4+32)=6−1(−x)−4(32)
Step 5.3.3.3
Multiply −1 by −1.
f(−x4+32)=6+1x−4(32)
Step 5.3.3.4
Multiply x by 1.
f(−x4+32)=6+x−4(32)
Step 5.3.3.5
Cancel the common factor of 2.
Step 5.3.3.5.1
Factor 2 out of −4.
f(−x4+32)=6+x+2(−2)(32)
Step 5.3.3.5.2
Cancel the common factor.
f(−x4+32)=6+x+2⋅(−2(32))
Step 5.3.3.5.3
Rewrite the expression.
f(−x4+32)=6+x−2⋅3
f(−x4+32)=6+x−2⋅3
Step 5.3.3.6
Multiply −2 by 3.
f(−x4+32)=6+x−6
f(−x4+32)=6+x−6
Step 5.3.4
Combine the opposite terms in 6+x−6.
Step 5.3.4.1
Subtract 6 from 6.
f(−x4+32)=x+0
Step 5.3.4.2
Add x and 0.
f(−x4+32)=x
f(−x4+32)=x
f(−x4+32)=x
Step 5.4
Since f−1(f(x))=x and f(f−1(x))=x, then f−1(x)=−x4+32 is the inverse of f(x)=6−4x.
f−1(x)=−x4+32
f−1(x)=−x4+32
