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 [tex] \sqrt[3]{x-3} [/tex]=y-1 add 1 [tex] \sqrt[3]{x-3} [/tex]+1=y switch y with g⁻¹(x) g⁻¹(x)=[tex] \sqrt[3]{x-3} [/tex]+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 [tex] \sqrt[3]{x-2} [/tex]=y replace with f⁻¹(x) f⁻¹(x)=[tex] \sqrt[3]{x-2} [/tex]
5. f(n)=5[tex] \sqrt{-n+1/2} [/tex] replace y=5[tex] \sqrt{-n+1/2} [/tex] switch n and y n=5[tex] \sqrt{-y+1/2} [/tex] solve for y n=5[tex] \sqrt{-y+1/2} [/tex] divide both sides by 5 n/5=n=[tex] \sqrt{-y+1/2} [/tex] square both sides [tex] \frac{n^2}{25} [/tex]=-y+1/2 subtract 1/2 from both sides [tex] \frac{n^2}{25} [/tex]-[tex] \frac{1}{2} [/tex]=-y multiply both sides by -1 -[tex] \frac{n^2}{25} [/tex]+[tex] \frac{1}{2} [/tex]=y replace with f⁻¹(n) f⁻¹(n)=-[tex] \frac{n^2}{25} [/tex]+[tex] \frac{1}{2} [/tex]
