Function Notation f(x)

Learn to read and use function notation to describe mathematical relationships.

Intermediate25 minLesson
Printable worksheet

Definition

Function notation is a way to name and describe functions using symbols like f(x)f(x), g(x)g(x), or h(x)h(x).
The notation f(x)f(x) is read as "f of x" and represents:
  • ff**: the name of the function
  • xx**: the input value (independent variable)
  • f(x)f(x)**: the output value (dependent variable)
For example, if f(x)=2x+3f(x) = 2x + 3, then:
f(5)=2(5)+3=10+3=13f(5) = 2(5) + 3 = 10 + 3 = 13
This means: when the input is 55, the output is 1313.

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What does f(x)f(x) mean?

Worked Examples

If f(x)=3x−2f(x) = 3x - 2, find f(4)f(4).
1

Identify what to substitute

f(4)f(4) means we replace xx with 44 → Input = 4
2

Substitute the value

f(4)=3(4)−2f(4) = 3(4) - 2
3

Multiply first

3×4=123 \times 4 = 12 → 12−212 - 2
4

Subtract

12−2=1012 - 2 = 10 → f(4)=10f(4) = 10

Common Mistakes

Thinking f(x)f(x) means ff multiplied by xx

Why it's wrong: The notation f(x)f(x) looks like multiplication, but it actually means "the function ff applied to the input xx." The parentheses indicate a function, not multiplication.

Correct: Read f(x)f(x) as "f of x": it represents the output when xx is the input.

Forgetting to substitute everywhere

Why it's wrong: When xx appears multiple times in a function, all instances must be replaced.

Correct: For f(x)=x2+2xf(x) = x^2 + 2x, to find f(3)f(3): replace BOTH xx's to get 32+2(3)=9+6=153^2 + 2(3) = 9 + 6 = 15.

Confusing the function name with the variable

Why it's wrong: Students sometimes mix up ff (the function name) with xx (the input variable).

Correct: ff names the rule; xx is what goes in. Different functions can use the same variable: f(x)=2xf(x) = 2x and g(x)=x+5g(x) = x + 5.

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Interactive Visual

Function graph

-4-2024-50510xyf(4)(4, 10)g(−3)(-3, 10)

Tap or hover over the graph to read off values.

f(x) = 3x − 2g(x) = x² + 1

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

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Try these examples:

3x - 2

m=3, b=-2

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Practice Problems

16 problems
Problem 1 of 16
Easy
What does f(x)f(x) mean?

Why It Matters

Function notation is essential in mathematics because:
  • Clear communication: Instead of saying "the function where you double and add 3," we simply write f(x)=2x+3f(x) = 2x + 3
  • Multiple functions: We can work with several functions at once: f(x)f(x), g(x)g(x), h(x)h(x)
  • Precise evaluation: f(4)f(4) clearly means "find the output when the input is 4"
  • Foundation for advanced math: Calculus, statistics, and computer science all rely on function notation
Function notation is used everywhere from physics formulas to smartphone apps!

Real World Applications

Temperature Conversion

Converting between Celsius and Fahrenheit uses function notation.

Example:

The function F(C)=95C+32F(C) = \frac{9}{5}C + 32 converts Celsius to Fahrenheit. So F(25)=95(25)+32=45+32=77°FF(25) = \frac{9}{5}(25) + 32 = 45 + 32 = 77°F.

1Try It Yourself

You need to convert 20°C to Fahrenheit.

What is F(20)F(20)?

Step 1: Write the mathematical expression

Substitute 20 into F(C)=95C+32F(C) = \frac{9}{5}C + 32:

Ride-Share Pricing

Ride-share apps calculate fares using functions.

Example:

If the fare is C(m)=2.50+1.75mC(m) = 2.50 + 1.75m where mm is miles, then a 6-mile ride costs C(6)=2.50+1.75(6)=2.50+10.50=13C(6) = 2.50 + 1.75(6) = 2.50 + 10.50 = 13 dollars.

2Try It Yourself

A taxi company charges a base fare plus a per-mile rate. The cost function is C(m)=3+2mC(m) = 3 + 2m dollars.

How much does a 10-mile trip cost?

Step 1: Write the mathematical expression

Find C(10)C(10):

Key Takeaways

  • 1f(x)f(x) is read as "f of x" and represents a function named ff with input xx
  • 2To evaluate a function, substitute the input value for every xx in the expression
  • 3f(3)=10f(3) = 10 means: when the input is 33, the output is 1010
  • 4Different function names (ff, gg, hh) let us work with multiple functions at once
  • 5Function notation appears throughout science, engineering, and technology

Frequently Asked Questions

Function names let us work with multiple relationships at once. Instead of saying "the first equation" and "the second equation," we can say f(x)f(x) and g(x)g(x). This is especially useful when comparing functions or combining them.
Function names let us work with multiple relationships at once. Instead of saying "the first equation" and "the second equation," we can say f(x)f(x) and g(x)g(x). This is especially useful when comparing functions or combining them.
Yes! While ff, gg, and hh are most common, any letter works. In science, meaningful letters are often used: v(t)v(t) for velocity as a function of time, A(r)A(r) for area as a function of radius.
Yes, they mean the same thing. f(x)f(x) represents the output of the function, which we often call yy. So f(x)=2x+1f(x) = 2x + 1 is equivalent to y=2x+1y = 2x + 1 when working with the function ff.

Glossary

Function notation
A way to name functions using symbols like f(x)f(x), read as "f of x"
Input
The value substituted into a function (the xx in f(x)f(x))
Output
The result of applying a function to an input (the value of f(x)f(x))
Evaluate
To find the output of a function for a specific input value
Function rule
The expression that defines how to calculate the output from the input
Today's challengeOne problem, three tries

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