Composite functions are an essential concept in algebra, particularly for Cambridge IGCSE students. They involve combining two functions such that the output of one function becomes the input of another. This process not only enhances problem-solving skills but also deepens the understanding of algebraic relationships and functions.

Introduction to Composite Functions

At the heart of composite functions is the operation of applying one function to the result of another. This is denoted as $g(f(x))$, signifying that $f(x)$ is computed first, followed by applying $g$ to the result of $f(x)$.

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• Key Concepts:
  • Function Composition: Combining two functions where the output of one becomes the input of another.
  • Notation: For two functions $f(x)$ and $g(x)$, the composite function $g \circ f$ is written as $g(f(x))$.

Understanding the Notation

The notation $g(f(x))$ is interpreted as "g of f of x," emphasizing the sequence of applying $f$ to $x$ and then $g$ to the outcome of $f(x)$.

Forming Composite Functions

To create a composite function, one must:

1. Determine the inner function.

2. Apply the result of this inner function to the outer function.

3. Simplify the resultant expression if possible.

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Example: Creating a Composite Function

Consider the functions $f(x) = \frac{3}{x} + 2$ and $g(x) = (3x + 5)^2$. Let's form the composite function $g(f(x))$.

Solution:

1. Compute $f(x)$: $f(x) = \frac{3}{x} + 2$

2. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(\frac{3}{x} + 2)$

3. Simplify the composite function:

After substitution and simplification, we find:

This process illustrates the method of forming and understanding composite functions through substitution and simplification.

Practice Problems

Problem: Composite Function Calculation

Given:

• $h(x) = x - 1$
• $j(x) = \sqrt{x + 2}$

Calculate $j(h(x))$.

Solution:

1. Calculate $h(x)$: $h(x) = x - 1$

2. Substitute $h(x)$ into $j(x)$: $j(h(x)$ =$\sqrt{(x - 1) + 2}$

3. Simplify: $j(h(x)) = \sqrt{x + 1}$

Worked Examples

Example 1: Basic Composite Function

Given functions:

• $f(x) = \frac{3}{x} + 2$
• $g(x) = (3x + 5)^2$

Form the composite function $g(f(x))$:

1. Substitute $f(x)$ into $g(x)$: $g(f(x)$= $\left[3\left(\frac3x+ 2\right) + 5\right]^2$

2. Simplify: $g(f(x)$ = $\left(\frac9x + 6 + 5\right)^2$= $\left(\frac9x + 11\right)^2$

3. Further simplify to make it clearer: $g(f(x))$= $\left(\frac{11x + 9}{x}\right)^2$

Example 2: Composite Function with Square Root

Given functions:

• $h(x) = x - 1$
• $j(x) = \sqrt{x + 2}$

Find $j(h(x))$:

1. Substitute $h(x)$ into $j(x)$: $j(h(x)$= $\sqrt{(x - 1) + 2}$

2. Simplify: $j(h(x)$ = $\sqrt{x + 1}$

Example 3: Composite Function with Reciprocal

Given functions:

• $p(x) = 3x - 4$
• $q(x) = \frac{1}{x}$

Calculate $q(p(x))$:

• Substitute $p(x)$ into $q(x)$: $q(p(x)$= $\frac{1}{3x - 4}$

Example 4: Composite Function with Quadratic and Reciprocal

Consider:

• $m(x) = x^2 + 2x + 1$
• $n(x) = \frac{1}{x + 1}$

Form $n(m(x))$:

1. Substitute $m(x)$ into $n(x)$: $n(m(x)$ = $\frac{1}{x^2 + 2x + 1 + 1}$

2. Simplify: $n(m(x)$ =$\frac{1}{x^2 + 2x + 2}$