To determine which statement must be true given the property [tex]\( f(-x) = f(x) \)[/tex] for all values of [tex]\( x \)[/tex], let's analyze the given property and the effects it has on the graph of [tex]\( f \)[/tex].
1. The property [tex]\( f(-x) = f(x) \)[/tex] tells us that for every [tex]\( x \)[/tex], [tex]\( f \)[/tex] takes the same value at both [tex]\( x \)[/tex] and [tex]\( -x \)[/tex]. This is the definition of an even function.
2. For a function [tex]\( f \)[/tex] to be even, its graph must be symmetric with respect to the [tex]\( y \)[/tex]-axis. This means if you reflect the graph across the [tex]\( y \)[/tex]-axis, it will look identical.
Let's now evaluate each given statement based on this property:
- Option A: The graph of [tex]\( f \)[/tex] is symmetric to the [tex]\( y \)[/tex]-axis.
Since we've concluded that an even function is symmetric with respect to the [tex]\( y \)[/tex]-axis, this statement is true.
- Option B: The graph of [tex]\( f \)[/tex] is symmetric to the [tex]\( x \)[/tex]-axis.
This statement is not necessarily true. A function is symmetric to the [tex]\( x \)[/tex]-axis if and only if for all [tex]\( x \)[/tex], [tex]\( f(x) = -f(-x) \)[/tex]. However, our given property doesn’t provide this condition, so this statement is false.
- Option C: The graph of [tex]\( f \)[/tex] is symmetric to the line [tex]\( y = x \)[/tex].
A function is symmetric to the line [tex]\( y = x \)[/tex] if [tex]\( f \)[/tex] is its own inverse, meaning [tex]\( f(f(x)) = x \)[/tex]. This does not relate directly to the property [tex]\( f(-x) = f(x) \)[/tex]. Thus, this statement is not necessarily true.
- Option D: [tex]\( f \)[/tex] is its own inverse function.
For [tex]\( f \)[/tex] to be its own inverse, we need [tex]\( f(f(x)) = x \)[/tex]. The given property [tex]\( f(-x) = f(x) \)[/tex] doesn't provide any information about [tex]\( f(f(x)) = x \)[/tex], so this statement is not necessarily true.
- Option E: [tex]\( f(-x) + f(x) = 0 \)[/tex] for all values of [tex]\( x \)[/tex].
This statement is claiming that [tex]\( f(-x) = -f(x) \)[/tex], which describes an odd function. Since [tex]\( f \)[/tex] is even as given by [tex]\( f(-x) = f(x) \)[/tex], this statement is false and contradictory to the property provided.
Therefore, the correct answer is:
A. The graph of [tex]\( f \)[/tex] is symmetric to the [tex]\( y \)[/tex]-axis.
