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.
• only the equation is a function.
• only the inverse is a function.
• both the equation and its inverse are functions.

Explanation:

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

Use the vertical line test. For any \( x \), \( f(x) \) has only one output (e.g., \( x = 2 \) gives \( 12 \), \( x = -2 \) also gives \( 12 \), but each \( x \) maps to one \( y \)). So \( f(x) \) is a function.

1. Find the inverse and check if it’s a function:

To find the inverse, swap \( x \) and \( y \): \( x = y^2 + 8 \), then solve for \( y \): \( y^2 = x - 8 \), so \( y = \pm\sqrt{x - 8} \).
For \( x > 8 \), each \( x \) gives two \( y \)-values (e.g., \( x = 12 \) gives \( y = 2 \) and \( y = -2 \)). By the vertical line test (or definition of a function: one input → one output), the inverse is not a function.

Thus, only the original equation is a function.

Answer:

Only the equation is a function.