Grade: Grade 9 Subject: Mathematics Unit: Functions & Modeling SAT: AdvancedMath ACT: Math

Function Notation

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Function

A function is a relation that assigns exactly one output value to each input value. We write f(x) = ... to define a function, where x is the input (independent variable) and f(x) is the output (dependent variable).

Function notation is a powerful way to describe mathematical relationships. Instead of writing y = 2x + 3, we write f(x) = 2x + 3. This notation makes it clear that we have a "function machine" that takes an input and produces an output.

Understanding Function Notation

Notation Meaning Example
f(x) The function f evaluated at x (the output when input is x) If f(x) = 2x, then f(3) = 6
f(3) The value of the function when x = 3 Substitute 3 for x and calculate
f(a + 1) The value of the function when x = a + 1 Substitute (a + 1) for x
f(x) = 5 Find x such that the output equals 5 Solve the equation for x

Reading Function Notation

f(x) is read as "f of x" - NOT "f times x." The parentheses indicate function evaluation, not multiplication. Think of it as: "the function f applied to the value x."

Evaluating Functions

To evaluate a function at a specific value:

  1. Take the function rule (e.g., f(x) = 3x - 2)
  2. Replace every x with the given input value
  3. Simplify to find the output

Domain and Range

Domain: The set of all possible input values (x-values) for a function.

Range: The set of all possible output values (y-values or f(x)-values) for a function.

SAT/ACT Connection

Function notation questions are extremely common on the SAT and ACT. You'll need to evaluate functions, find inputs given outputs, and interpret function notation in context. Master the basics, and these become easy points!

Examples

Work through these examples to master function notation.

Example 1: Basic Evaluation

Problem: If f(x) = 4x - 7, find f(5).

Step 1: Replace x with 5 in the function rule

f(5) = 4(5) - 7

Step 2: Simplify

f(5) = 20 - 7 = 13

Answer: f(5) = 13

Example 2: Evaluation with Negative Input

Problem: If g(x) = x² + 3x, find g(-2).

Step 1: Replace x with -2

g(-2) = (-2)² + 3(-2)

Step 2: Simplify carefully with negatives

g(-2) = 4 + (-6) = 4 - 6 = -2

Answer: g(-2) = -2

Example 3: Evaluation with an Expression

Problem: If f(x) = 2x + 1, find f(a + 3).

Step 1: Replace x with (a + 3)

f(a + 3) = 2(a + 3) + 1

Step 2: Distribute and simplify

f(a + 3) = 2a + 6 + 1 = 2a + 7

Answer: f(a + 3) = 2a + 7

Example 4: Finding Input Given Output

Problem: If f(x) = 3x - 5, find x when f(x) = 10.

Step 1: Set up the equation

3x - 5 = 10

Step 2: Solve for x

3x = 15

x = 5

Answer: x = 5 (Check: f(5) = 3(5) - 5 = 15 - 5 = 10)

Example 5: Using a Table

Problem: Given the table, find f(2) and find x if f(x) = 7.

x01234
f(x)13579

Find f(2): Look in the table where x = 2

f(2) = 5

Find x if f(x) = 7: Look for 7 in the f(x) row

When f(x) = 7, x = 3

Practice

Evaluate the functions and select the correct answer.

1. If f(x) = 5x + 2, find f(4).

A) 18   B) 22   C) 27   D) 20

2. If g(x) = x² - 4, find g(3).

A) 5   B) 9   C) -1   D) 13

3. If h(x) = 2x - 9, find h(-3).

A) -15   B) -3   C) 3   D) -6

4. If f(x) = x² + 2x, find f(-4).

A) 8   B) 24   C) -8   D) 0

5. If f(x) = 3x + 1, find f(2a).

A) 6a + 1   B) 6a + 2   C) 3a + 2   D) 5a + 1

6. If g(x) = 4x - 3 and g(x) = 17, find x.

A) 3   B) 4   C) 5   D) 6

7. If f(x) = x² - 1 and f(x) = 24, find x (positive value).

A) 4   B) 5   C) 6   D) 7

8. If h(x) = 2x + 5, what is h(x + 1)?

A) 2x + 6   B) 2x + 7   C) 2x + 8   D) 2x + 5

9. If f(3) = 11 and f(x) = 2x + b, find b.

A) 3   B) 4   C) 5   D) 6

10. If f(x) = -x + 8, for what value of x does f(x) = x?

A) 2   B) 4   C) 6   D) 8

Click to reveal answers
  1. B) 22 - f(4) = 5(4) + 2 = 20 + 2 = 22
  2. A) 5 - g(3) = 3² - 4 = 9 - 4 = 5
  3. A) -15 - h(-3) = 2(-3) - 9 = -6 - 9 = -15
  4. A) 8 - f(-4) = (-4)² + 2(-4) = 16 - 8 = 8
  5. A) 6a + 1 - f(2a) = 3(2a) + 1 = 6a + 1
  6. C) 5 - 4x - 3 = 17; 4x = 20; x = 5
  7. B) 5 - x² - 1 = 24; x² = 25; x = 5 (positive)
  8. B) 2x + 7 - h(x + 1) = 2(x + 1) + 5 = 2x + 2 + 5 = 2x + 7
  9. C) 5 - 2(3) + b = 11; 6 + b = 11; b = 5
  10. B) 4 - -x + 8 = x; 8 = 2x; x = 4

Check Your Understanding

Answer these reflection questions to deepen your understanding.

1. What is the difference between f(x) and f times x?

Reveal Answer

f(x) represents function notation - it means "the function f evaluated at x" or "the output of f when the input is x." It is NOT multiplication. The parentheses indicate function evaluation. In contrast, f times x would be written as f · x or fx (if f were a variable), which represents multiplication. This is a common source of confusion, so always remember: f(x) = "f of x," not "f times x."

2. Explain how to find x when given f(x) = some value.

Reveal Answer

When you're given f(x) = some value (like f(x) = 10), you need to find the input that produces that output. Set the function rule equal to the given value and solve for x. For example, if f(x) = 2x + 3 and f(x) = 11, then 2x + 3 = 11, so 2x = 8, and x = 4. This is the reverse of evaluation: instead of finding output from input, you're finding input from output.

3. Why does function notation use letters like f, g, h instead of just y?

Reveal Answer

Function notation allows us to work with multiple functions in the same problem without confusion. With y notation, if we have two relationships, we'd need y₁ and y₂, which becomes cumbersome. With function notation, we can write f(x), g(x), and h(x) clearly. It also emphasizes that the output depends on the input, and makes it easier to evaluate functions at specific values (writing f(5) is cleaner than "y when x = 5").

4. What does it mean for a relation to be a function?

Reveal Answer

A relation is a function if and only if each input has exactly ONE output. In other words, you can't have the same x-value paired with two different y-values. Graphically, this is the "vertical line test": if any vertical line crosses the graph more than once, it's not a function. For example, y = x² is a function (each x gives one y), but x = y² is not (x = 4 gives both y = 2 and y = -2).

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review