Exercise 125 Page 98

Practice makes perfect

a Some functions are categorized as even and odd functions. These categories are defined as follows.

Even functions:& f(- x)=f(x) Odd functions:& f(- x)=- f(x)

Even functions

The equation f(- x)=f(x) tells us that if we substitute the opposite of an input, an even function will give the same output for both inputs. |c|c| Input & Output x & f(x) - x & f(- x)

This means that all functions that exhibit a line of reflection along the y-axis are even functions. For example, all monomial functions with an even degree, will model this behaviour. Let's show this for y=x^2.

Odd functions

The equation f(- x)=- f(x) tells us that if we substitute the opposite of an input into a function, if the function is odd it will give the opposite output of f(x). |c|c| Input & Output x & - f(x) - x & f(- x) All monomial functions with an odd degree will model this behaviour. Let's show this for y=x^3.

Given function

Let's plot the given function and determine the output for two inputs that are opposite numbers. If the outputs are not the same it's not an even function. If the outputs are not opposite numbers, it is not an odd function.

Since the outputs of 1 and -1 are neither the same nor opposite, this is neither even or odd. f(- x) &≠ f(x) * f(- x) &≠ - f(x) *

b Again, let's graph the function and pick two opposite inputs and check their outputs.

Since the outputs of 1 and -1 are neither the same nor opposite, this is neither even or odd. f(- x) &≠ f(x) * f(- x) &≠ - f(x) *

c Like in previous parts, we will graph the function and check the outputs for two inputs that are opposite numbers.

In this case, the outputs are the same which suggests that this is an even function. If we draw a vertical line along x=0, we notice that what's on the left-hand side of the line is the mirror image of what is on the right-hand side.