Chapter 3, Problem 1SGA

To determine

To find: the function even or odd which is symmetric to the $y$ -axis.

Answer to Problem 1SGA

An even function is symmetric with respect to the $y$ -axis.

Explanation of Solution

Given:

Even function and odd function.

Concept used:

An even function is symmetric with respect to the $y$ -axis because:

$\left(-a,b\right)\in S$ if and only if $\left(a,b\right)\in S$ .

In the elimination method or substitution method:

Either add or subtract equation to get an equation in one variable.

When the coefficients in one variable are opposites then add the equations to eliminate the variable and when the coefficients of one variable are equal and subtract the equation to eliminate a variable.

Calculation:

Let’s consider an odd and even function is a symmetric with respect to the $y$ -axis.

Need to find whether an odd function or even function is a symmetric with respect to the $y$ -axis.

An even function is symmetric with respect to the $y$ -axis because:

$\left(-a,b\right)\in S$ if and only if $\left(a,b\right)\in S$ .

Test: substituting $\left(a,b\right)\text{ and }\left(-a,b\right)$ into the equation produces equivalent equations.

Example:

Substituting $\left(2,8\right)\text{ and }\left(-2,8\right)$ in the equation $y=-x^2+12$ produces now substituting $\left(2,8\right)$ :

  $\begin{array}{l}y=-x^2+12.\\ 8=-{\left(2\right)}^2+12.\\ 8=8.\end{array}$

Now substituting $\left(-2,8\right)$ :

  $\begin{array}{l}y=-x^2+12.\\ \Rightarrow 8=-{\left(-2\right)}^2+12.\\ \Rightarrow 8=8.\end{array}$

Hence, an even function is symmetric with respect to the $y$ -axis.