Functions and graphs are a significant component of the ACT® Math section. Understanding these concepts can improve your performance on the test, as they appear frequently throughout the exam.
Functions that Follow the Pattern of an Equation
These show up all the time on the ACT® Math section. Many students don't know how to do these because they're not taught in the standard course of math classes in high school. They're not hard at all.
You'll likely get questions on this topic in the first 10 questions based on difficulty. Those will be pretty basic. Then, you'll get more in questions 20-60 that will follow the same rules, except have more steps to them.
Function Notation
Function notation is a method of expressing mathematical relationships. When you encounter an expression like f(x) = x + 1:
- f(x) represents the function
- x is the input variable
- The expression after the equals sign (x + 1) defines the operation performed on the input
For example, to evaluate f(3):
f(3) = 3 + 1 = 4
Approach to Solving Function Problems
- Identify the function definition
- Determine the input value
- Substitute the input value for each x in the function
- Calculate the result
Practice Section
Question 1:
For f(g(x)) where g(x) = 2x and f(x) = x + 3, what is f(g(4))?
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Solution:
g(4) = 2 * 4 = 8
f(g(4)) = f(8) = 8 + 3 = 11
Therefore, f(g(4)) = 11
Question 2:
For f(x, y) = x + y, what is f(4, 3)?
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Solution:
f(4, 3) = 4 + 3 = 7
Therefore, f(4, 3) = 7
Question 3:
For -f(g(x)) where g(x) = x - 1 and f(x) = 2x + 1, what is -f(g(5))?
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Solution:
g(5) = 5 - 1 = 4
f(4) = 2 * 4 + 1 = 9
-f(g(5)) = -9
Therefore, -f(g(5)) = -9
Key Points to Remember
- Function notation expresses mathematical relationships concisely
- Always substitute the given input value for x in the function
- Solve problems methodically, following each step carefully
Regular practice with functions and graphs will improve your proficiency and speed on the ACT® Math section.