Functions and Graphs
2.5 Operations with Functions and Symmetries
Combining Functions
Function operations use outputs just like number operations.
Definition: Function Operations
Given $f$ and $g$:
- $(f+g)(x)=f(x)+g(x)$
- $(f-g)(x)=f(x)-g(x)$
- $(fg)(x)=f(x)g(x)$
- $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$, where $g(x)\ne0$
Example: Add and divide functions
Let $f(x)=x+1$ and $g(x)=x-2$.
Find $(f+g)(x)$ and $\left(\frac{f}{g}\right)(x)$.
Show Solution
Add:
$$(f+g)(x)=(x+1)+(x-2)=2x-1.$$
Divide:
$$\left(\frac{f}{g}\right)(x)=\frac{x+1}{x-2},\quad x\ne2.$$
MyOpenMath: Practice: function operations (placeholder)
Even and Odd Symmetry
Symmetry can simplify graphing and equation checks.
Definition: Even and Odd Functions
- $f$ is even if $f(-x)=f(x)$ for all $x$ in the domain.
- $f$ is odd if $f(-x)=-f(x)$ for all $x$ in the domain.
Fact: Symmetry meaning
- Even functions are symmetric about the y-axis.
- Odd functions are symmetric about the origin.
Example: Classify symmetry
Classify each function.
- $f(x)=x^2+4$
- $g(x)=x^3-2x$
Show Solution
- $f(-x)=(-x)^2+4=x^2+4=f(x)$, so $f$ is even.
- $g(-x)=(-x)^3-2(-x)=-x^3+2x=-(x^3-2x)=-g(x)$, so $g$ is odd.