Evaluate a function with a sum or difference as the input such as [latex]f(x+1)[/latex].
Evaluate the sum, difference, product, or quotient of two functions at given inputs.
Find the sum, difference, product, or quotient of two functions and state the domain.
Just as we have performed algebraic operations on polynomials, we can do the same with polynomial functions.
Evaluate Polynomial Functions
You can evaluate polynomial functions similar to how you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.
Example
Given the function [latex]f(x)=3x^{2}-2x+1[/latex]:
1) Find [latex]f(-1)[/latex]
2) Find [latex]f(x+1)[/latex]
Show Solution
1)
[latex]\begin{align} \require{color}f({\color{Green}{-1}})&=3({\color{Green}{-1}})^{2}-2({\color{Green}{-1}})+1 &&\color{blue}{\textsf{substitute -1 for each x in the polynomial}}\\ &= 3(1)-2(-1)+1 &&\color{blue}{\textsf{follow order of operations, evaluate exponents first, then multiply}}\\ &= 3+2+1 &&\color{blue}{\textsf{add}}\\ &= 6 \end{align}[/latex]
2)
[latex]\begin{align} \require{color}f({\color{Green}{x+1}})&=3({\color{Green}{x+1}})^{2}-2({\color{Green}{x+1}})+1 &&\color{blue}{\textsf{substitute x+1 for each x in the polynomial}}\\ &= 3(x+1)(x+1)-2(x+1)+1 &&\color{blue}{\textsf{follow order of operations}}\\ &= 3(x^2+2x+1)-2x-2+1\\ &= 3x^2+6x+3-2x-2+1 &&\color{blue}{\textsf{combine like terms}}\\ &= 3x^2+4x+2 \end{align}[/latex]
In the following video, we show more examples of evaluating polynomials for given values of the variable.
In this section, we will focus on how to perform algebraic operations on polynomial functions and introduce related notation.
The four basic operations on functions are adding, subtracting, multiplying, and dividing. If [latex]f[/latex] and [latex]g[/latex] are functions and [latex]x[/latex] is in the domain of both functions, then:
We will focus on applying these operations to polynomial functions in this section.
Operations such as addition, subtraction, multiplication, and division can be used to produce a new function from two or more functions. The domain of the new function will be the intersection of the domains of the initial functions.
Add and Subtract Polynomial Functions
Adding and subtracting polynomial functions is the same as adding and subtracting polynomials. When you evaluate a sum or difference of functions, you can either evaluate first or perform the operation on the functions first as we will see. Our next examples describe the notation used to add and subtract polynomial functions.
ExAMPLE
Let [latex]f(x)=3x^2-6x-2[/latex] and [latex]g(x)=4x-1[/latex]
Compute the following:
[latex]1)\, f(-1)+g(4)\\[/latex]
[latex]2)\, f(-1)-g(4)\\[/latex]
[latex]3)\, g(-2)-g(3)\\[/latex]
Show Solution
1)
[latex]f(-1)+g(4)[/latex]
To find [latex]f(-1),[/latex] substitute [latex]\require{color}\color{Green}{-1}[/latex] in place of [latex]x[/latex] in the given function. To find [latex]f(4)[/latex] substitute [latex]\require{color}\color{Green}{4}[/latex] in place of [latex]x[/latex] in the given function.
To find [latex]g(-2),[/latex] substitute [latex]\require{color}\color{Green}{-2}[/latex] in place of [latex]x[/latex] in the given function. To find [latex]g(3)[/latex] substitute [latex]\require{color}\color{Green}{3}[/latex] in place of [latex]x[/latex] in the given function.
The domain of the combined function is the intersection of the domains of the original functions.
The domain of [latex]f(x)=2x^3-5x+3[/latex] is [latex](-\infty, \infty)[/latex] because there are no restrictions on the domain.
The domain of [latex]h(x)=x-5[/latex] is [latex](-\infty, \infty)[/latex] because there are also no restrictions on the domain.
The intersection of the two domains will also be [latex](-\infty, \infty)[/latex].
The domain of the combined function is the intersection of the domains of the original functions.
The domain of [latex]f(x)=2x^3-5x+3[/latex] is [latex](-\infty, \infty)[/latex] because there are no restrictions on the domain.
The domain of [latex]h(x)=x-5[/latex] is [latex](-\infty, \infty)[/latex] because there are also no restrictions on the domain.
The intersection of the two domains will also be [latex](-\infty, \infty)[/latex].
In our next example, we will evaluate a sum and difference of functions and show that you can get to the same result in one of two ways.
Example
Let [latex]f(x)=2x^3-5x+3[/latex] and [latex]h(x)=x-5[/latex]
Evaluate: [latex](f+h)(2)[/latex]
Show that you get the same result by
1) Evaluating the functions first, then performing the indicated operation on the result.
2) Performing the operation on the functions first then evaluating the result.
Show Solution
1) First, we will evaluate the functions separately:
Both methods give the same result, and both require about the same amount of work.
ExAMPLE
Given: [latex]f(x)=x^2-3x\\[/latex]
Find:
[latex]1)f(x+1)-f(2)\\[/latex]
[latex]2)f(x+1)+f(x)\\[/latex]
[latex]3)f(x+h)-f(x)\\[/latex]
Show Solution
[latex]1)f(x+1)-f(2)\\[/latex]
To find [latex]f(x+1),[/latex] substitute [latex]\require{color}\color{Green}{x+1}[/latex] in place of [latex]x[/latex] in the given function. To find [latex]f(2)[/latex] substitute [latex]\require{color}\color{Green}{2}[/latex] in place of [latex]x[/latex] in the given function.
We could take the same steps as we did above by finding [latex](f \cdot g)(t)[/latex] and then substituting [latex]-1[/latex] in place of [latex]t[/latex]. Here is another way you can evaluate [latex](f \cdot g)(-1)[/latex].
[latex](f \cdot g)(-1)= f(-1) \cdot g(-1)[/latex]
Let’s first find [latex]f(-1)[/latex] and [latex]g(-1)[/latex]. Then multiply the results together.
Domain: All real numbers except [latex]x\not=-3.[/latex]The value [latex]x=-3[/latex] will make the denominator equal to zero and therefore the expression will be undefined.[latex]\\[/latex]
Domain in Interval Notation: [latex](-\infty, -3)\bigcup(-3, \infty)\\[/latex]
Domain in Set-Builder Notation: [latex]\{ x | x\not=-3 \}\\\\[/latex]
In our next example we will find function values from a graph.
EXAMPLE
For the functions [latex]\color{DarkRed}{f(x)}[/latex], red graph, and [latex]\color{blue}{g(x)}[/latex], blue graph, pictured, find the function values.
[latex]1)f(-1)[/latex]
[latex]2)(f-g)(2)[/latex]
[latex]3)(f+g)(0)[/latex]
[latex]4)\left(\dfrac{f}{g}\right)(2)\\[/latex]
Show Solution
1)[latex]\begin{align}\require{color}f(-1)=-2 &&\color{blue}{\textsf{when the input is -1, the output is -2}} \end{align}\\[/latex]