Function (Basics)

Table of Contents
1. Function Basics
2. What Is a Function?
3. Function Notation
4. Evaluating Functions
5. Solving for Input Values
6. Composite Function Evaluation
7. Functions with Multiple Operations
8. Common Mistakes
9. Strategies and Tips
10. Practice Questions
11. Answer Key and Solutions
View more

Function Basics

Functions are one of the most fundamental concepts in algebra and appear frequently on the SSAT Upper Level test. A function is a special relationship between two sets of numbers where each input value corresponds to exactly one output value. Understanding how to work with functions, evaluate them, and interpret their notation is essential for success on test day. This topic builds the foundation for more advanced algebraic reasoning and problem-solving skills.

What Is a Function?

A function is a rule that assigns to each input exactly one output. We often use the notation \( f(x) \) to represent a function named \( f \) with input \( x \). The value \( f(x) \) is called the output or the value of the function at \( x \).

For example, if \( f(x) = 2x + 3 \), then the function \( f \) takes an input \( x \), multiplies it by 2, and then adds 3. The result is the output.

The domain of a function is the set of all possible input values. The range is the set of all possible output values.

Function Notation

Function notation is a compact way to express the relationship between inputs and outputs. When you see \( f(x) \), read it as "f of x" or "the value of f at x."

If \( f(x) = x^2 - 5 \), then:

\( f(3) \) means substitute 3 for \( x \):
\( f(3) = 3^2 - 5 \)
\( f(3) = 9 - 5 \)
\( f(3) = 4 \)

Similarly, \( f(-2) \) means substitute -2 for \( x \):
\( f(-2) = (-2)^2 - 5 \)
\( f(-2) = 4 - 5 \)
\( f(-2) = -1 \)

You can also evaluate functions at expressions. For instance, \( f(a + 1) \) means substitute \( a + 1 \) for every occurrence of \( x \):
\( f(a + 1) = (a + 1)^2 - 5 \)
\( f(a + 1) = a^2 + 2a + 1 - 5 \)
\( f(a + 1) = a^2 + 2a - 4 \)

Evaluating Functions

To evaluate a function at a specific value, follow these steps:

1. Identify the function rule.
2. Substitute the given input value for the variable in the function.
3. Simplify the expression using the order of operations.

Worked Example 1

Problem: If \( g(x) = 3x - 7 \), what is \( g(5) \)?

Solution:
Substitute 5 for \( x \):
\( g(5) = 3(5) - 7 \)
\( g(5) = 15 - 7 \)
\( g(5) = 8 \)

Answer: 8

Worked Example 2

Problem: If \( h(x) = x^2 + 2x - 1 \), what is \( h(-3) \)?

Solution:
Substitute -3 for \( x \):
\( h(-3) = (-3)^2 + 2(-3) - 1 \)
\( h(-3) = 9 - 6 - 1 \)
\( h(-3) = 2 \)

Answer: 2

Worked Example 3

Problem: If \( f(x) = \frac{x + 4}{2} \), what is the value of \( f(10) \)?

1. 3
2. 5
3. 7
4. 9
5. 14

Solution:
Substitute 10 for \( x \):
\( f(10) = \frac{10 + 4}{2} \)
\( f(10) = \frac{14}{2} \)
\( f(10) = 7 \)

Answer: C

Solving for Input Values

Sometimes you are given the output of a function and asked to find the input. This requires solving an equation.

Worked Example 4

Problem: If \( f(x) = 4x + 5 \) and \( f(a) = 21 \), what is the value of \( a \)?

Solution:
Set up the equation:
\( 4a + 5 = 21 \)
Subtract 5 from both sides:
\( 4a = 16 \)
Divide both sides by 4:
\( a = 4 \)

Answer: 4

Worked Example 5

Problem: If \( g(x) = x^2 - 3 \) and \( g(b) = 13 \), what is a possible value of \( b \)?

1. -4
2. -3
3. 3
4. 4
5. 10

Solution:
Set up the equation:
\( b^2 - 3 = 13 \)
Add 3 to both sides:
\( b^2 = 16 \)
Take the square root of both sides:
\( b = 4 \) or \( b = -4 \)

Both 4 and -4 are solutions. Looking at the answer choices, choice D gives 4, which is a correct value.

Answer: D

Composite Function Evaluation

A composite function involves applying one function to the result of another. If you see \( f(g(x)) \), this means evaluate \( g(x) \) first, then use that result as the input for \( f \).

Worked Example 6

Problem: If \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), what is \( f(g(5)) \)?

Solution:
First, evaluate \( g(5) \):
\( g(5) = 5 - 3 = 2 \)

Now substitute this result into \( f \):
\( f(g(5)) = f(2) \)
\( f(2) = 2(2) + 1 \)
\( f(2) = 4 + 1 \)
\( f(2) = 5 \)

Answer: 5

Worked Example 7

Problem: If \( h(x) = x^2 \) and \( k(x) = x + 2 \), what is \( h(k(3)) \)?

1. 5
2. 9
3. 11
4. 25
5. 36

Solution:
First, evaluate \( k(3) \):
\( k(3) = 3 + 2 = 5 \)

Now substitute this result into \( h \):
\( h(k(3)) = h(5) \)
\( h(5) = 5^2 \)
\( h(5) = 25 \)

Answer: D

Functions with Multiple Operations

Some functions involve several operations. Always follow the order of operations: parentheses, exponents, multiplication and division from left to right, then addition and subtraction from left to right.

Worked Example 8

Problem: If \( f(x) = 3(x - 2)^2 + 5 \), what is \( f(4) \)?

Solution:
Substitute 4 for \( x \):
\( f(4) = 3(4 - 2)^2 + 5 \)
\( f(4) = 3(2)^2 + 5 \)
\( f(4) = 3(4) + 5 \)
\( f(4) = 12 + 5 \)
\( f(4) = 17 \)

Answer: 17

Common Mistakes

Mistake 1: Ignoring Negative Signs

When substituting a negative number into a function, students often forget to use parentheses, leading to sign errors.

Incorrect: If \( f(x) = x^2 + 3x \) and you evaluate \( f(-2) \) as \( -2^2 + 3(-2) = -4 - 6 = -10 \).

Correct: \( f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2 \).

Always use parentheses around negative numbers when substituting.

Mistake 2: Confusing \( f(x + a) \) with \( f(x) + a \)

These are not the same. \( f(x + a) \) means substitute \( x + a \) into the function. \( f(x) + a \) means evaluate the function at \( x \) and then add \( a \).

For example, if \( f(x) = x^2 \):

\( f(x + 1) = (x + 1)^2 = x^2 + 2x + 1 \)

\( f(x) + 1 = x^2 + 1 \)

These are different expressions.

Mistake 3: Order of Operations Errors

When evaluating functions with multiple operations, failing to follow the correct order can lead to wrong answers. Always apply exponents before multiplication, and multiplication before addition or subtraction.

If \( g(x) = 2x^2 - 5 \) and you want \( g(3) \):

Incorrect: \( 2 \times 3^2 - 5 = 6^2 - 5 = 36 - 5 = 31 \)

Correct: \( 2 \times 3^2 - 5 = 2 \times 9 - 5 = 18 - 5 = 13 \)

Strategies and Tips

Tip 1: Write Out Each Step

Even if the problem seems simple, write out the substitution and simplification steps. This reduces careless errors and helps you track your work under timed conditions.

Tip 2: Use Parentheses Liberally

When substituting values, especially negative numbers or expressions, use parentheses to avoid sign and order of operation errors.

Tip 3: Check Your Answer

If you have time, plug your answer back into the original function to verify it produces the expected result. This is especially useful for problems where you solve for an input.

Tip 4: Recognize Common Function Forms

Linear functions have the form \( f(x) = mx + b \). Quadratic functions have the form \( f(x) = ax^2 + bx + c \). Knowing these patterns helps you anticipate the structure of the problem.

Practice Questions

Question 1

If \( f(x) = 5x - 8 \), what is \( f(6) \)?

1. 14
2. 22
3. 28
4. 30
5. 38

Question 2

If \( g(x) = x^2 - 4x + 7 \), what is \( g(3) \)?

1. 2
2. 4
3. 6
4. 10
5. 16

Question 3

If \( h(x) = \frac{2x + 6}{4} \), what is \( h(9) \)?

1. 3
2. 4
3. 5
4. 6
5. 7

Question 4

If \( f(x) = 3x + 2 \) and \( f(a) = 17 \), what is the value of \( a \)?

1. 3
2. 4
3. 5
4. 6
5. 7

Question 5

If \( k(x) = x^2 + 1 \) and \( m(x) = 2x \), what is \( k(m(3)) \)?

1. 7
2. 13
3. 19
4. 28
5. 37

Question 6

If \( f(x) = 4 - 2x \), what is \( f(-5) \)?

1. -14
2. -6
3. 6
4. 14
5. 18

Question 7

If \( g(x) = (x + 3)^2 \), what is \( g(-1) \)?

1. 1
2. 2
3. 4
4. 8
5. 16

Question 8

If \( p(x) = 7 - x \) and \( p(b) = 3 \), what is the value of \( b \)?

1. 2
2. 3
3. 4
4. 5
5. 10

Question 9

If \( f(x) = 2x^2 - 3x + 1 \), what is \( f(2) \)?

1. 1
2. 3
3. 5
4. 7
5. 9

Question 10

If \( h(x) = x + 5 \) and \( j(x) = 3x - 1 \), what is \( j(h(2)) \)?

1. 12
2. 15
3. 18
4. 20
5. 21

Answer Key and Solutions

Question 1: B

Substitute 6 for \( x \):
\( f(6) = 5(6) - 8 \)
\( f(6) = 30 - 8 \)
\( f(6) = 22 \)

Question 2: B

Substitute 3 for \( x \):
\( g(3) = 3^2 - 4(3) + 7 \)
\( g(3) = 9 - 12 + 7 \)
\( g(3) = 4 \)

Question 3: D

Substitute 9 for \( x \):
\( h(9) = \frac{2(9) + 6}{4} \)
\( h(9) = \frac{18 + 6}{4} \)
\( h(9) = \frac{24}{4} \)
\( h(9) = 6 \)

Question 4: C

Set up the equation:
\( 3a + 2 = 17 \)
Subtract 2 from both sides:
\( 3a = 15 \)
Divide both sides by 3:
\( a = 5 \)

Question 5: E

First, evaluate \( m(3) \):
\( m(3) = 2(3) = 6 \)

Now substitute this result into \( k \):
\( k(m(3)) = k(6) \)
\( k(6) = 6^2 + 1 \)
\( k(6) = 36 + 1 \)
\( k(6) = 37 \)

Question 6: D

Substitute -5 for \( x \):
\( f(-5) = 4 - 2(-5) \)
\( f(-5) = 4 + 10 \)
\( f(-5) = 14 \)

Question 7: C

Substitute -1 for \( x \):
\( g(-1) = (-1 + 3)^2 \)
\( g(-1) = (2)^2 \)
\( g(-1) = 4 \)

Question 8: C

Set up the equation:
\( 7 - b = 3 \)
Subtract 7 from both sides:
\( -b = -4 \)
Multiply both sides by -1:
\( b = 4 \)

Question 9: B

Substitute 2 for \( x \):
\( f(2) = 2(2)^2 - 3(2) + 1 \)
\( f(2) = 2(4) - 6 + 1 \)
\( f(2) = 8 - 6 + 1 \)
\( f(2) = 3 \)

Question 10: D

First, evaluate \( h(2) \):
\( h(2) = 2 + 5 = 7 \)

Now substitute this result into \( j \):
\( j(h(2)) = j(7) \)
\( j(7) = 3(7) - 1 \)
\( j(7) = 21 - 1 \)
\( j(7) = 20 \)