1.1 Review of functions  (Page 9/28)

In contrast, looking at [link] again, if a function $f$ is symmetric about the origin, then whenever the point $(x,y)$ is on the graph, the point $\left(\text{−}x,\text{−}y\right)$ is also on the graph. In other words, $f(\text{−}x)=\text{−}f(x).$ If $f$ has this property, we say $f$ is an odd function, which has symmetry about the origin. For example, $f(x)=x^3$ is odd because

One symmetric function that arises frequently is the absolute value function    , written as $\left|x\right|.$ The absolute value function is defined as

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if $x<0,$ then $\left|x\right|=\text{−}x>0,$ and if $x>0,$ then $\left|x\right|=x>0.$ However, for $x=0,\left|x\right|=0.$ Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if $x=0,$ the output $\left|x\right|=0.$ We conclude that the range of the absolute value function is $\left\{y\right|y\ge 0\}.$ In [link] , we see that the absolute value function is symmetric about the y -axis and is therefore an even function.

Key concepts

• A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
• If no domain is stated for a function $y=f\left(x\right),$ the domain is considered to be the set of all real numbers $x$ for which the function is defined.
• When sketching the graph of a function $f,$ each vertical line may intersect the graph, at most, once.
• A function may have any number of zeros, but it has, at most, one y -intercept.
• To define the composition $g\circ f,$ the range of $f$ must be contained in the domain of $g.$
• Even functions are symmetric about the $y$ -axis whereas odd functions are symmetric about the origin.