(a) A function f is said to be even if the equation...

(a) A function $f$ is said to be even if the equation $f(-x)=f(x)$ is satisfied by all values of $x$ in the domain of $f .$ Explain why the graph of an even function must be symmetric about the $y$ -axis. (b) Show that each function is even by computing $f(-x)$ and then noting that $f(x)$ and $f(-x)$ are equal. (i) $f(x)=x^{2}$ (iii) $f(x)=3 x^{6}-\frac{4}{x^{2}}+1$ (ii) $f(x)=2 x^{4}-6$

(a) A function $f$ is said to be even if the equation $f(-x)=f(x)$ is satisfied by all values of $x$ in the domain of $f .$ Explain why the graph of an even function must be symmetric about the $y$ -axis.
(b) Show that each function is even by computing $f(-x)$ and then noting that $f(x)$ and $f(-x)$ are equal.
(i) $f(x)=x^{2}$
(iii) $f(x)=3 x^{6}-\frac{4}{x^{2}}+1$
(ii) $f(x)=2 x^{4}-6$

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Key Concepts

Graph Symmetry about the y?axis

The graph of an even function is symmetric about the y?axis because for every point (x, f(x)) on the graph, the point (–x, f(–x)) is also present. Given that f(–x) equals f(x) for even functions, the corresponding y?value at x and –x is the same, thereby creating a mirror image on either side of the y?axis. This symmetry is a fundamental visual characteristic used to identify even functions.

Verification by Substitution

One standard method to verify that a function is even is to compute f(–x) and compare it to f(x). If these two expressions are identical for all x within the function's domain, then the function satisfies the definition of being even. This technique is commonly used in examples and proofs involving even functions by applying algebraic manipulation to establish equality.

Even Functions

An even function is defined by the property f(-x) = f(x) for every x in its domain. This definition captures the idea that the function treats positive and negative inputs symmetrically. In algebra, this concept plays an important role in understanding the structure of functions and their graphs, often facilitating simplifications in computations and analyses.

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