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Introduction to Functions

1. What is a Function?

A function is a rule that assigns exactly one output (\( y \)) for each input (\( x \)).

  • ✔ Every input has only one output.
  • ✔ If you plug in the same input, you always get the same result.

Example:

If a bus ticket price depends on the distance traveled, it is a function.
If a person's age depends on their birth year, it is a function.
If a machine sometimes gives two different outputs for the same input, it is not a function.

2. Function Notation

Functions are often written as: \( f(x) = \text{expression} \).
\( f(x) \) means "function of \( x \)" (similar to \( y \)). Here, \( x \) is the input and \( f(x) \) is the output.

Example:
If \( f(x) = x + 5 \), then \( f(2) = 2 + 5 = 7 \).
If \( g(x) = 2x - 3 \), then \( g(4) = 2(4) - 3 = 5 \).

3. Domain and Range

Domain: The set of all possible input values (\( x \)-values).
Range: The set of all possible output values (\( y \)-values).

Example: If \( f(x) = x^2 \) and the domain is \(\{0,1,2,3\}\), then the range is \(\{0,1,4,9\}\) because \(0^2 = 0\), \(1^2 = 1\), \(2^2 = 4\), and \(3^2 = 9\).
If a function accepts all real numbers, its domain is \((-\infty, \infty)\).

4. The Vertical Line Test

A graph represents a function if no vertical line crosses it more than once.

Example:
\( y = x^2 \) passes the vertical line test (it is a function), whereas \( x^2 + y^2 = 9 \) fails the test (it represents a circle, not a function).

5. Graphing a Function

To graph a function, plug in values for \( x \) and find the corresponding \( y \).

Example: Graph \( f(x) = x + 2 \):
\( f(0) = 0 + 2 = 2 \) → (0,2)
\( f(1) = 1 + 2 = 3 \) → (1,3)
\( f(2) = 2 + 2 = 4 \) → (2,4)

If the points form a straight line, it's a linear function; if they curve, it’s nonlinear.

6. Finding the Inverse of a Function

The inverse of a function undoes the original function. If \( f(x) \) turns \( x \) into \( y \), then the inverse function \( f^{-1}(x) \) turns \( y \) back into \( x \).

  1. Replace \( f(x) \) with \( y \). For example, \( y = 3x + 2 \).
  2. Swap \( x \) and \( y \): \( x = 3y + 2 \).
  3. Solve for \( y \): \( y = \frac{x - 2}{3} \).
  4. Replace \( y \) with \( f^{-1}(x) \): \( f^{-1}(x) = \frac{x - 2}{3} \).

Example: Find the inverse of \( f(x) = 2x - 5 \).
Swap \( x \) and \( y \): \( x = 2y - 5 \).
Solve for \( y \): \( y = \frac{x + 5}{2} \).
So, \( f^{-1}(x) = \frac{x + 5}{2} \).

7. Real-Life Applications of Functions

  • ✔ Physics: Speed depends on time traveled.
  • ✔ Banking: Interest grows as a function of time.
  • ✔ Temperature Conversion: \( F = 1.8C + 32 \).
  • ✔ Business: Profit depends on items sold.

Practice Questions 🎯

  1. If \( f(x) = x + 5 \), find \( f(3) \).
  2. What is the domain of \( f(x) = x^2 \) for \( x = 0, 1, 2, 3 \)?
  3. Is \( y = x^2 \) a function? Why?
  4. What is \( f(6) \) if \( f(x) = 2x - 1 \)?
  5. Graph \( f(x) = x + 1 \) for \( x = -2, -1, 0, 1, 2 \).
  6. If \( g(x) = 3x - 4 \), what is \( g(5) \)?
  7. Determine whether the set of points \( (4,2), (5,3), (6,2), (7,8) \) represents a function.
  8. Find the inverse of \( f(x) = x + 3 \).
  9. Apply the vertical line test to determine if \( x^2 + y^2 = 9 \) is a function.
  10. Find the inverse of \( f(x) = 5x - 7 \).

Want a Bigger Challenge? 🤔🔥

  1. Find \( f^{-1}(x) \) if \( f(x) = \frac{2x + 3}{5} \).
  2. Solve for \( x \) if \( f(x) = 8 \) for \( f(x) = 3x - 2 \).
  3. Graph the inverse of \( f(x) = x + 4 \).
  4. Find \( g(x) \) if \( f(x) = 2x + 1 \) and \( g(x) = 3f(x) - 2 \).
  5. If a company’s revenue follows \( R(x) = 100x - 500 \), find \( x \) when revenue is \$1500.

10. Summary

  • ✅ Functions assign exactly one output per input.
  • ✅ Inverse functions undo each other.
  • ✅ Functions are used in real-world applications like physics, finance, and business.
  • ✅ Domain and range are key to understanding a function's behavior.

11. Congratulations

🎉 Congratulations! You have mastered Level 1 Algebra! 🚀

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