Question

Which function must have a restricted domain in order to have an inverse function?

a. f(x)= 6x -10
b. f(x) =8x^2-6x+10
c. f(x) =3(x-2)^3+1
d. f(x)= √x+3​

Answer 1

B

Explanation:

Consider the function
`f(x)=8x^2-6x+10`

Rewrite it as

`f(x)\\ \\=8\left(x^2-(3)/(4)x+(5)/(4)\right)\\ \\=8\left(x^2-2\cdot (3)/(8)\cdot x+(5)/(4)\right)\\ \\=8\left(x^2-2\cdot (3)/(8)\cdot x+(9)/(64)-(9)/(64)+(5)/(4)\right)\\ \\=8\left(x-(3)/(8)\right)^2+142`

The domain of this function is
`x\in (-\infty, \infty)` and the range is
`y\in [142,\infty)`

Find the inverse function:

`y-142=8\left(x-(3)/(8)\right)^2\\ \\x=\pm \sqrt{(1)/(8)(y-142)}+(3)/(8)\\ \\f^(-1)(x)= \sqrt{(1)/(8)(x-142)}+(3)/(8)`

So, the domain of the inverse function is
`x\in [142,\infty)` and the range will be
`\left[(3)/(8),\infty\right)`

So, the domain must be restricted