Table of contents  
What is a function?  
How do I evaluate functions?  
How do I evaluate composite functions?  
How do I compose functions? 
A function takes an input and produces an output. In function notation, f(x), f is the name of the function, a is the input variable, and f(x) is the output.
For example, given f(x) = 2x + 1, the expression 2x + 1 works as instructions on what to do with the input x. In this case, the input x is multiplied by 2, then 1 is added to the product.
Evaluating functions algebraically and using tables
When we encounter an algebraic function, we can find the value of the function at specific inputs. For example, for f(x) = 2x + 1, we can calculate f(2), the output of the function f when its input, x, is equal to 2.
To evaluate a function at a specific input value:
Example: If f(x) = x^{3} 9, what is the value of f(2)?
Sol: We need to plug2 into f for x to evaluate f(2):
f(x) = x^{3 }9
f(2) = (2)^{3}9
=89
= 17
17 is the value of f(2).
With algebraic functions, we can evaluate the function using multiple inputs to create multiple inputoutput pairs. These inputoutput pairs can be put in a table, as shown below for f(x) = 2x + 1.
Sometimes, a table of inputoutput pairs is provided without an algebraic function. Consider the table below.
The table contains four inputoutput pairs. We can interpret the information in the table as:
To evaluate a function using a table:
Example:
Based on the table above, what is the value of f (3)?
Sol: In the table, the inputs are in the x column, and the outputs are in the f(x) column.
The value of f(3) is in the same row as the xvalue of 3, so f(3) = 5.
5 is the value of f(3).
Evaluating composite functions algebraically and using tables
A composite function uses the output of one function as the input of another. For example, for f(g(x)):
As such, composite functions should be worked from the inside out. Order matters when evaluating composite functions: g(f(x)) is not the same as f(g(x))!
For example, for f(x) = x^{2} and g(x) = x + 1, f(g(1)) = 4, but g(f(1)) = 2.
To evaluate composite functions at a specific input value:
Example: If f(x) = 2x + 1 and g(x) = x^{2 }2x+1, what is the value of f(g(1))?
Sol: We need to input 1 into g, then use the output as the input for f.
First, let's evaluate g(1):
g(x) = x^{2}2x+1
g(1)=(1)^{2}2(1)+1
=1+2+1
= 4
Next, we need to input 4 into f to evaluate f(4):
f(x) = 2x + 1
f(4) = 2(4) +1
= 8+1
= 9
9 is the value of f(g(1)).
Composite functions can also be evaluated using a table. The table can have an additional column for a total of three: one column for input and two columns for the outputs of two functions. Consider the table for f(x) = x^{2} and g(x) = x + 1:
From the table, we can tell that g(1) = 2, and f(2) = 4. Therefore, f(g(1)) = 4.
To evaluate composite functions at a specific input value given a table:
Example:
The table above provides the values of functions f and g at several values of x. What is the value of g(f(2))?
Sol: We need to identify the value of f for x = 2, then use that value as the value of x to identify the value of g.
According to the table, when x = 2, f(x) = 5. When x = 5, g(x) = 16:
g(f(2)) = g(5) = 16
16 is the value of g(f(2)).
Inputting expressions instead of values into functions
In addition to inputting a specific value, we can also input one function into another function, which creates a composite function defined by a single expression.
For example, for f(x) = x^{2} and g(x) = x + 1, f(g(x)) replaces each instance of x in f with g(x), which is equal to x + 1: f(g(x)) = (x + 1)^{2}. Inputting an expression into a function, e.g., f(x + 1), works similarly.
A function can also be defined in terms of another function. For example, for f(x) = x^{2} and g(x) = f(x) + 1, we can replace the f(x) in function g with 22: g(x) = x^{2} + 1.
If you find yourself struggling to rewrite complex functions, you might want to brush up on the Operations with polynomials and Operations with rational expressions skills, which have their own lessons.
To compose two functions:
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