Question

1. which of the following is true for the relation $f(x) = x^2 + 8$?

• neither the equation nor its inverse is a function.
• both the equation and its inverse are functions.
• only the equation is a function.
• only the inverse is a function.

Explanation:

Step1: Check if \( f(x) = x^2 + 8 \) is a function.

A relation is a function if for every input \( x \), there is exactly one output \( y \). For \( f(x) = x^2 + 8 \), no matter what real number \( x \) we choose, squaring it and adding 8 gives exactly one result. So \( f(x) = x^2 + 8 \) is a function (it passes the vertical line test).

Step2: Find the inverse of \( f(x) = x^2 + 8 \) and check if it's a function.

To find the inverse, first, let \( y = x^2 + 8 \). Then, solve for \( x \):

$$
LATEXBLOCK0
$$

Now, swap \( x \) and \( y \) to get the inverse relation: \( y = \pm\sqrt{x - 8} \). For a given \( x > 8 \), there are two possible \( y \)-values (one positive and one negative square root). So the inverse relation does not pass the vertical line test (it's a parabola opening to the right, and a vertical line will intersect it at two points), meaning the inverse is not a function.

So only the original equation \( f(x) = x^2 + 8 \) is a function.

Answer:

Only the equation is a function.