If f is an even function, determine whether g is even, odd, or neither. Explain. (a) g(x)=

step 1 We're told that f isn't even, and we're given function G in terms of F, and we're asked to deter

If f is an even function, determine whether g is even, odd,...

Key Concepts

Even Functions

An even function is defined by the property f(x) = f(?x) for all x in its domain. This symmetry about the y-axis means that the function’s values mirror on either side of the vertical axis, and any transformation that preserves this mirror symmetry will keep the function even.

Odd Functions

An odd function satisfies the condition f(?x) = –f(x) for every x. This implies symmetry with respect to the origin, meaning that rotating the graph 180° about the origin leaves it unchanged. Understanding odd functions is important when analyzing how different operations affect the function's symmetry.

Function Transformations

Function transformations involve operations such as scaling, reflecting, and translating the original function, and each can affect the symmetry properties differently. Multiplying the function by a constant (like –1) may preserve evenness or oddness when done uniformly, while horizontal shifts tend to disrupt even symmetry because they move the center of symmetry away from the y-axis.

Vertical Translation

Vertical translation, which involves adding or subtracting a constant to a function, shifts the entire graph up or down without altering the function's horizontal symmetry. Thus, if the original function is even, subtracting or adding a constant will result in another even function.

Horizontal Translation

Horizontal translation shifts the graph of a function left or right. For functions that are even, such a shift generally destroys the y-axis symmetry since the reflection of a horizontally shifted function does not coincide with the original graph unless corrected by an equivalent shift.

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