Step-by-Step Guidance for Composition and Decomposition of Functions

Q1. Let $f(x) = x^2 - 3x$ and $g(x) = 4x - 1$. Find:

• (a) $(f \circ g)(x)$

• (b) $(g \circ f)(x)$

Background

Topic: Composition of Functions

This question tests your understanding of how to compose two functions, meaning how to substitute one function into another.

Key Terms and Formulas

• Composition: $(f \circ g)(x) = f(g(x))$

• Domain: The set of $x$ values for which the composition is defined.

Step-by-Step Guidance

1. For $(f \circ g)(x)$, start by finding $g(x)$, which is $4x - 1$.

2. Substitute $g(x)$ into $f(x)$: $f(g(x)) = (g(x))^2 - 3g(x)$.

3. Write out the expression: $f(g(x)) = (4x - 1)^2 - 3(4x - 1)$.

4. Expand $(4x - 1)^2$ and $-3(4x - 1)$ separately.

5. Combine like terms to simplify the expression, but stop before the final simplification.

Try solving on your own before revealing the answer!

Final Answer:

$(f \circ g)(x) = 16x^2 - 8x + 1 - 12x + 3 = 16x^2 - 20x + 4$

We substituted $g(x)$ into $f(x)$ and simplified the result.

Q2. Let $f(x) = \frac{2}{4-x}$ and $g(x) = x^2 + 1$. Find:

• (a) $(f \circ g)(x)$

• (b) $(f \circ g)(3)$

• (c) $(f \circ f)(2)$

• (d) Domain of $g \circ f$

Background

Topic: Composition of Functions and Domain Analysis

This question tests your ability to compose functions, evaluate compositions at specific values, and determine the domain of a composite function.

Key Terms and Formulas

• Composition: $(f \circ g)(x) = f(g(x))$

• Domain: Values of $x$ for which the function is defined (no division by zero, etc.)

Step-by-Step Guidance

1. For $(f \circ g)(x)$, first find $g(x) = x^2 + 1$.

2. Substitute $g(x)$ into $f(x)$: $f(g(x)) = \frac{2}{4 - g(x)}$.

3. Write the expression: $f(g(x)) = \frac{2}{4 - (x^2 + 1)}$.

4. Simplify the denominator: $4 - x^2 - 1$.

5. For $(f \circ g)(3)$, substitute $x = 3$ into the simplified expression, but stop before calculating the final value.

Try solving on your own before revealing the answer!

Final Answer:

$(f \circ g)(x) = \frac{2}{3 - x^2}$

For $(f \circ g)(3)$, substitute $x = 3$ to get $\frac{2}{3 - 9} = \frac{2}{-6}$.

Q3. Let $f(x) = 3x - 5$ and $g(x) = \sqrt{2x + 1}$. Find $(f \circ g)(x)$ and state the domain.

Background

Topic: Composition of Functions and Domain Analysis

This question tests your ability to compose a linear function with a square root function and determine the domain of the composite function.

Key Terms and Formulas

• Composition: $(f \circ g)(x) = f(g(x))$

• Domain: For $g(x) = \sqrt{2x + 1}$, $2x + 1 \geq 0$.

Step-by-Step Guidance

1. Find $g(x) = \sqrt{2x + 1}$.

2. Substitute $g(x)$ into $f(x)$: $f(g(x)) = 3g(x) - 5$.

3. Write the expression: $f(g(x)) = 3\sqrt{2x + 1} - 5$.

4. Determine the domain: Set $2x + 1 \geq 0$ and solve for $x$.

5. Express the domain in interval notation, but stop before stating the exact interval.

Try solving on your own before revealing the answer!

Final Answer:

$(f \circ g)(x) = 3\sqrt{2x + 1} - 5$

Domain: $x \geq -\frac{1}{2}$

Q4. Decompose the following functions: Find $f(x)$ and $g(x)$ such that $h(x) = (f \circ g)(x)$.

• (a) $h(x) = \frac{4}{\sqrt{2x - 5}}$

• (b) $h(x) = 2 + (3x + 1)^4$

Background

Topic: Decomposition of Functions

This question tests your ability to break a function into two simpler functions whose composition gives the original function.

Key Terms and Formulas

• Decomposition: $h(x) = (f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. For (a), identify an inner function $g(x)$ that is inside another operation (e.g., $\sqrt{2x - 5}$).

2. Let $g(x) = \sqrt{2x - 5}$, then $f(x) = \frac{4}{x}$ so that $f(g(x)) = \frac{4}{\sqrt{2x - 5}}$.

3. For (b), identify the inner function $g(x) = 3x + 1$, and the outer function $f(x) = 2 + x^4$.

4. Check that $f(g(x))$ matches $h(x)$ for each case.

Try solving on your own before revealing the answer!

Final Answer:

(a) $f(x) = \frac{4}{x}$, $g(x) = \sqrt{2x - 5}$

(b) $f(x) = 2 + x^4$, $g(x) = 3x + 1$