Question

if ( f ) is a function with the property that ( f(-x) = f(x) ) for all values of ( x ), which statement must be true?

a. the graph of ( f ) is symmetric to the ( y )-axis.

b. the graph of ( f ) is symmetric to the ( x )-axis.

c. the graph of ( f ) is symmetric to the line ( y = x ).

d. ( f ) is its own inverse function.

e. ( f(-x) + f(x) = 0 ) for all values of ( x ).

Explanation:

A function \( f \) with \( f(-x)=f(x) \) for all \( x \) is an even function. By the definition of even functions, their graphs are symmetric about the \( y \)-axis. Option B is incorrect as function graphs (except for the zero function in a trivial sense) aren't symmetric about the \( x \)-axis (it would not pass the vertical line test). Option C is for inverse functions (symmetric about \( y = x \)), not even functions. Option D: An even function being its own inverse would require \( f(f(x))=x \), which isn't true for most even functions (e.g., \( f(x)=x^2 \), \( f(f(x))=(x^2)^2=x^4
eq x \) generally). Option E: \( f(-x)+f(x)=0 \) is for odd functions (\( f(-x)= -f(x) \)), not even functions. So only option A is correct.

Answer:

A. The graph of \( f \) is symmetric to the \( y \)-axis.