Question

Find the inverse of each function

1) g (x)= 3+(x-1)^3
4) f (x)=x^3+2
5) f (n)= 5 square root of (-n+1/2)

Answer 1

to find the inverse
1. replace f(x) or g(x) or f(n) with y
2. switch places of x and y
3. solve for y
4. replace y with f⁻¹(x) or g⁻¹(x) or f⁻¹(n)



1.
g(x)=3+(x-1)^3
replace
y=3+(x-1)^3
switch
x=3+(y-1)^3
solve
x=3+(y-1)^3
subtract 3
x-3=(y-1)^3
cube root both sides

`\sqrt[3]{x-3}`=y-1
add 1

`\sqrt[3]{x-3}`+1=y
switch y with g⁻¹(x)
g⁻¹(x)=
`\sqrt[3]{x-3}`+1





4. f(x)=x^3+2
replace with y
y=x^3+2
switch x and y
x=y^3+2
solve for y
x=y^3+2
subtract 2
x-2=y^3
cube roo both sides

`\sqrt[3]{x-2}`=y
replace with f⁻¹(x)
f⁻¹(x)=
`\sqrt[3]{x-2}`



5.
f(n)=5
`√(-n+1/2)`
replace
y=5
`√(-n+1/2)`
switch n and y
n=5
`√(-y+1/2)`
solve for y
n=5
`√(-y+1/2)`
divide both sides by 5
n/5=n=
`√(-y+1/2)`
square both sides

`(n^2)/(25)`=-y+1/2
subtract 1/2 from both sides

`(n^2)/(25)`-
`(1)/(2)`=-y
multiply both sides by -1
-
`(n^2)/(25)`+
`(1)/(2)`=y
replace with f⁻¹(n)
f⁻¹(n)=-
`(n^2)/(25)`+
`(1)/(2)`



1. g⁻¹(x)=
`\sqrt[3]{x-3}`+1
4. f⁻¹(x)=
`\sqrt[3]{x-2}`
5. f⁻¹(n)=-
`(n^2)/(25)`+
`(1)/(2)`