Answer:
The functions are inverses; f(g(x)) = x ⇒ answer D
[tex]h^{-1}(x)=\sqrt{\frac{x+1}{3}}[/tex] ⇒ answer D
Step-by-step explanation:
* Lets explain how to find the inverse of a function
- Let f(x) = y
- Exchange x and y
- Solve to find the new y
- The new y = [tex]f^{-1}(x)[/tex]
* Lets use these steps to solve the problems
∵ [tex]f(x)=\sqrt{x-3}[/tex]
∵ f(x) = y
∴ [tex]y=\sqrt{x-3}[/tex]
- Exchange x and y
∴ [tex]x=\sqrt{y-3}[/tex]
- Square the two sides
∴ x² = y - 3
- Add 3 to both sides
∴ x² + 3 = y
- Change y by [tex]f^{-1}(x)[/tex]
∴ [tex]f^{-1}(x)=x^{2}+3[/tex]
∵ g(x) = x² + 3
∴ [tex]f^{-1}(x)=g(x)[/tex]
∴ The functions are inverses to each other
* Now lets find f(g(x))
- To find f(g(x)) substitute x in f(x) by g(x)
∵ [tex]f(x)=\sqrt{x-3}[/tex]
∵ g(x) = x² + 3
∴ [tex]f(g(x))=\sqrt{(x^{2}+3)-3}=\sqrt{x^{2}+3-3}=\sqrt{x^{2}}=x[/tex]
∴ f(g(x)) = x
∴ The functions are inverses; f(g(x)) = x
* Lets find the inverse of h(x)
∵ h(x) = 3x² - 1 where x ≥ 0
- Let h(x) = y
∴ y = 3x² - 1
- Exchange x and y
∴ x = 3y² - 1
- Add 1 to both sides
∴ x + 1 = 3y²
- Divide both sides by 3
∴ [tex]\frac{x + 1}{3}=y^{2}[/tex]
- Take √ for both sides
∴ ± [tex]\sqrt{\frac{x+1}{3}}=y[/tex]
∵ x ≥ 0
∴ We will chose the positive value of the square root
∴ [tex]\sqrt{\frac{x+1}{3}}=y[/tex]
- replace y by [tex]h^{-1}(x)[/tex]
∴ [tex]h^{-1}(x)=\sqrt{\frac{x+1}{3}}[/tex]
