Functions and Graphs

Linear Functions

Linear functions are among the simplest types of functions, with the general form:

f(x) = mx + b

Learn more about Linear Functions

where:

  • m is the slope, representing the rate of change or how steep the line is.
  • b is the y-intercept, the point where the line crosses the y-axis.

A linear function creates a straight line when graphed, reflecting a constant rate of change. The graph of a linear function can slope up, down, or be horizontal, depending on the value of m.

Applications of Linear Functions

See examples of Linear Function applications

Linear functions are used in many real-world situations, such as:

  • Distance vs. Time: A car traveling at a constant speed illustrates a linear relationship between distance and time.
  • Cost Modeling: In economics, linear functions model cost as a function of quantity.
  • Temperature Conversion: Converting between Celsius and Fahrenheit is a linear relationship.

Interactive Example

Adjust the values for the slope (m) and y-intercept (b) to see how they affect the linear function graph:

Trigonometric Functions

Interactive Trigonometric Function Plot

Logarithmic Functions

Rational Functions

Polynomial Functions

Quadratic Functions

Understanding Exponential Functions

Exponential functions are mathematical functions of the form:

f(x) = a * b^x

Where:





Inverse Functions

The inverse of a function reverses its input-output relationship. For a function \( f(x) \), its inverse \( f^{-1}(x) \) satisfies \( f(f^{-1}(x)) = x \). Only one-to-one functions (which pass the horizontal line test) have inverses.

Steps to Find the Inverse of a Function:

  1. Replace \( f(x) \) with \( y \).
  2. Swap \( x \) and \( y \).
  3. Solve for \( y \) to get the inverse \( f^{-1}(x) \).

Interactive Graph

Understanding Exponential Functions

Exponential functions are mathematical functions of the form:

f(x) = a * b^x

Where:

Domain and Range

The **domain** of an exponential function is all real numbers (\( -\infty < x < \infty \)) because you can input any real number for \(x\).

The **range** depends on the value of \(a\) and \(b\):

Intercepts

The **y-intercept** of an exponential function occurs when \(x = 0\). Substituting \(x = 0\) into the equation, we get \( f(0) = a \cdot b^0 = a \), which is the value of the y-intercept.

The **x-intercept** (if any) is found by setting \( f(x) = 0 \), but for an exponential function with \(a > 0\) and \(b > 0\), there are no x-intercepts because the graph never crosses the x-axis.

Asymptotes

An **asymptote** is a line that the graph approaches but never reaches. For an exponential function, there is a **horizontal asymptote** at \( y = 0 \). This means that as \( x \) increases or decreases, the value of the function approaches 0 but never touches it.

Inverse Functions

The inverse of an exponential function is a **logarithmic function**. If the original function is \(f(x) = a \cdot b^x\), its inverse is \( f^{-1}(x) = \log_b(x/a) \). This means that exponential growth and decay functions can be "reversed" by their corresponding logarithmic functions.





Inverse Functions

The inverse of a function reverses its input-output relationship. For a function \( f(x) \), its inverse \( f^{-1}(x) \) satisfies \( f(f^{-1}(x)) = x \). Only one-to-one functions (which pass the horizontal line test) have inverses.

Steps to Find the Inverse of a Function:

  1. Replace \( f(x) \) with \( y \).
  2. Swap \( x \) and \( y \).
  3. Solve for \( y \) to get the inverse \( f^{-1}(x) \).

Interactive Graph