CodeMathFusion

Functions and Their Properties

1️⃣ Understanding Functions

A function is a special type of mathematical rule that assigns exactly one output for each input.

  • If you input a number into a function, you get only one result.
  • Every function can be written as \( f(x) \), where:
    • \( x \) is the input (or independent variable).
    • \( f(x) \) is the output (or dependent variable).

Example: Consider the function:

\[ f(x) = 2x + 3 \]

  • If \( x = 1 \), then \( f(1) = 2(1) + 3 = 5 \).
  • If \( x = -2 \), then \( f(-2) = 2(-2) + 3 = -1 \).

A function ensures that for every \( x \), there is only one \( f(x) \) (no multiple outputs for a single input).

Not a function:

\( y^2 = x \) (because one \( x \)-value can have two \( y \)-values).

2️⃣ Ways to Represent a Function

A function can be represented in multiple ways:

  1. Equation Form: Example: \( f(x) = 3x - 5 \).
  2. Table Form: Lists input-output pairs.
    \( x \) \( f(x) \)
    0 -5
    1 -2
    2 1
  3. Graph Form: A visual representation by plotting points.
  4. Verbal Description: For example, “A function that doubles the input and adds 3.”

3️⃣ The Vertical Line Test

A function’s graph must pass the vertical line test:

  • If a vertical line touches the graph at only one point everywhere, it is a function.
  • If a vertical line crosses the graph more than once, it is not a function.

Example:

  • \( y = x^2 \) ✅ Function (each \( x \) gives one \( y \)).
  • \( x = y^2 \) ❌ Not a function (some \( x \)-values yield two \( y \)-values).

4️⃣ Types of Functions

1️⃣ Linear Functions

A straight-line function in the form:

\[ f(x) = mx + b \]

- \( m \) is the slope and \( b \) is the y-intercept.

Example: \( f(x) = 2x + 3 \) (slope = 2, y-intercept = 3).

2️⃣ Quadratic Functions

Forms a parabola:

\[ f(x) = ax^2 + bx + c \]

Example: \( f(x) = x^2 - 4x + 3 \) (a U-shaped curve with a vertex and axis of symmetry).

3️⃣ Absolute Value Functions

Form a V-shape:

\[ f(x) = |x| \]

Example: \( f(2) = 2 \), \( f(-3) = 3 \).

4️⃣ Piecewise Functions

Defined by different rules for different intervals of \( x \):

Example: \[ f(x)= \begin{cases} x+2, & x < 0 \\ x^2, & x \ge 0 \end{cases} \]
For \( x = -1 \), \( f(-1)=1 \); for \( x = 2 \), \( f(2)=4 \).

5️⃣ Domain and Range

Domain (Input Values)

The set of all possible \( x \)-values that can be used as input.

Example: For \( f(x)=\sqrt{x} \), the domain is \( x\ge 0 \).

Range (Output Values)

The set of all possible \( f(x) \)-values (outputs).

Example: For \( f(x)=x^2 \), the range is \( y\ge 0 \).

Practice Questions – Fundamental

  1. Determine whether the following are functions: \( y = x^3+2x-1 \), \( x = y^2 \), \( y=|x| \).
  2. Find the domain and range of: \( f(x)=2x-3 \) and \( g(x)=\sqrt{x+4} \).
  3. Which of the following pass the vertical line test?
  4. Identify whether the following functions are linear, quadratic, or absolute value: \( y=4x-7 \), \( y=x^2-6x+5 \), \( y=|x-2| \).
  5. Find \( f(3) \) if \( f(x)=5x-2 \).
  6. If \( f(x)=x^2-2x+1 \), find \( f(-2) \).
  7. Graph the function \( f(x)=|x-3| \).
  8. Identify the slope and y-intercept of \( f(x)=-3x+5 \).
  9. Evaluate \( g(x)=3x^2-2x+4 \) for \( x=-1 \).
  10. Write a piecewise function that represents: \( f(x)=x+2 \) for \( x<0 \) and \( f(x)=x^2 \) for \( x\ge0 \).

Challenging Problems – Take It Up a Notch!

  1. Given \( f(x)=\frac{3x+2}{x-1} \), determine its domain.
  2. Find all \( x \) such that \( f(x)=g(x) \) for \( f(x)=2x+1 \) and \( g(x)=x^2-3 \).
  3. A company’s revenue is \( R(x)=500x-x^2 \). Find the maximum revenue.
  4. Find the equation of a function that passes through \( (2,5) \) and \( (4,9) \) with a slope of 2.
  5. The population of a city follows \( P(t)=5000+200t \) (where \( t \) is years since 2000). Find the population in 2030.

7️⃣ Summary

  • ✅ A function gives one output for each input.
  • ✅ The vertical line test determines whether a graph represents a function.
  • ✅ Functions can be represented as equations, tables, graphs, or verbally.
  • ✅ Domain and range define the possible inputs and outputs of a function.

8️⃣ Great Work!

Fantastic work! You're ready for the next challenge: More Advanced Function Topics!

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