Function Composition

Function composition involves combining two or more functions where the output of one function becomes the input of the next. For example, if f(x) = 2x + 3 and g(x) = x², then the composition (f ∘ g)(x) means applying g first, then f to the result: f(g(x)) = 2(x²) + 3.

Steps to Compose Functions:

• Identify the inner and outer functions.
• Substitute the output of the inner function into the outer function.

Interactive Practice Problem:

If f(x) = x + 5 and g(x) = 3x, find both (f ∘ g)(x) and (g ∘ f)(x). Show Solution

Inverse Functions

An inverse function reverses the effect of the original function. For example, if f(x) = 2x + 4, the inverse function is f⁻¹(x) = (x - 4)/2.

Finding Inverses:

1. Swap x and y in the equation.
2. Solve for y to find the inverse.

Interactive Problem: Try finding the inverse of f(x) = 3x - 7. Show Solution

Piecewise Functions

Piecewise functions are defined by different expressions based on the input. For example, f(x) = {x² for x ≤ 0, 2x + 1 for x > 0} means one rule applies for negative inputs, and another for positive inputs.

Understanding Domain Boundaries: Each segment of a piecewise function is valid for a specific part of the domain.

Example: Graph the function f(x) = {3x for x < 2, x² for x ≥ 2} to see how each piece behaves.

Domain and Range

The domain of a function is the set of all possible inputs (x-values), and the range is the set of all possible outputs (f(x)-values). For example, for f(x) = √x, the domain is x ≥ 0 and the range is y ≥ 0.

How to Find Domain and Range:

• Look for restrictions (e.g., division by zero, square roots).
• Identify possible values of x and corresponding values of y.

Interactive Problem: Determine the domain and range of f(x) = 1/(x - 3). Show Solution

Vertical Line Test

The vertical line test helps determine if a curve represents a function. If a vertical line intersects the graph at more than one point, the graph is not a function because a function can only map each input to one output.

Application of Vertical Line Test: Move a vertical line across the graph; if it touches more than once at any point, it is not a function.

Practice Activity: Test y = x² and x = y² to see which one passes the vertical line test. Show Solution

Function Transformations

Function transformations include shifting, reflecting, stretching, or compressing the graph of a function. For example, shifting f(x) = x² by 2 units to the right gives f(x - 2) = (x - 2)².

Types of Transformations:

• Vertical Shifts: Adding/subtracting a constant.
• Horizontal Shifts: Adding/subtracting inside the function.
• Reflections: Multiplying by -1.
• Stretches/Compressions: Scaling by a constant factor.

Practice Problem: Describe the transformation from f(x) = x² to g(x) = -2(x - 3)² + 4. Show Solution

Function Composition

Function composition allows us to combine two functions, where the output of one function becomes the input for another. For example, if f(x) = 2x + 3 and g(x) = x², then (f ∘ g)(x) is the same as f(g(x)).

Inverse Functions

An inverse function reverses the operation of the original function. If f(x) takes x to y, then f⁻¹(y) takes y back to x. For example, if f(x) = 3x + 4, the inverse function is f⁻¹(x) = (x - 4) / 3.

Piecewise Functions

Piecewise functions are defined by different rules depending on the input value. For instance, f(x) = {x² for x ≤ 0; 2x + 1 for x > 0} applies different expressions for different values of x.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of possible output values. For example, in f(x) = √x, the domain is x ≥ 0, and the range is y ≥ 0.

Vertical Line Test

The vertical line test determines if a graph represents a function. If a vertical line intersects the graph at more than one point, it’s not a function, as it means multiple outputs for a single input.

Function Transformations

Transformations include shifting, reflecting, stretching, or compressing a graph. For example, shifting f(x) = x² by 2 units right results in f(x - 2) = (x - 2)².