Function Composition Calculator

About Function Composition Calculator

Welcome to our Function Composition Calculator, a free online tool that helps you calculate the composition of two functions with detailed step-by-step instructions. Whether you are a student learning about function composition, preparing for calculus, or a teacher creating examples, this calculator provides clear explanations of the algebraic process.

What is Function Composition?

Function composition is the process of combining two functions to create a new function. When we compose functions f and g, we write it as $(f \circ g)(x)$, which is read as "f composed with g" or "f of g of x".

The notation $(f \circ g)(x)$ means $f(g(x))$, where:

• First, we apply g to the input x, getting $g(x)$
• Then, we apply f to that result, getting $f(g(x))$
• The inner function is applied first, then the outer function

How to Calculate Function Composition

To find $(f \circ g)(x) = f(g(x))$, follow these steps:

Step 1: Identify the Inner and Outer Functions

In $(f \circ g)(x)$, g is the inner function (applied first) and f is the outer function (applied second).

Step 2: Substitute g(x) into f(x)

Replace every occurrence of x in f(x) with the entire expression for g(x).

Step 3: Simplify

Expand, combine like terms, factor, or otherwise simplify the resulting expression.

Step 4: Write the Final Answer

Express your result as $(f \circ g)(x) = $ simplified expression.

Important Properties of Function Composition

Function Composition is NOT Commutative

In general, $(f \circ g)(x) \neq (g \circ f)(x)$. The order matters! This is one of the most important properties to remember.

Function Composition is Associative

If you have three functions f, g, and h, then $f \circ (g \circ h) = (f \circ g) \circ h$.

Identity Function

The identity function $I(x) = x$ satisfies $(f \circ I)(x) = (I \circ f)(x) = f(x)$ for any function f.

Inverse Functions

If f and g are inverse functions, then $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$.

Common Examples of Function Composition

$f(x)$	$g(x)$	$(f \circ g)(x) = f(g(x))$
$f(x) = 2x + 1$	$g(x) = x^2$	$2x^2 + 1$
$f(x) = x^2$	$g(x) = 2x + 1$	$(2x + 1)^2 = 4x^2 + 4x + 1$
$f(x) = \sqrt{x}$	$g(x) = x + 4$	$\sqrt{x + 4}$
$f(x) = e^x$	$g(x) = \ln(x)$	$e^{\ln(x)} = x$
$f(x) = \ln(x)$	$g(x) = e^x$	$\ln(e^x) = x$
$f(x) = \frac{1}{x}$	$g(x) = x + 2$	$\frac{1}{x + 2}$

Domain of Composite Functions

The domain of $(f \circ g)(x)$ consists of all x in the domain of g such that $g(x)$ is in the domain of f.

For example, if $f(x) = \sqrt{x}$ and $g(x) = x - 4$:

• $g(x) = x - 4$ is defined for all real numbers
• $f(x) = \sqrt{x}$ requires $x \geq 0$
• For $(f \circ g)(x) = \sqrt{x - 4}$, we need $x - 4 \geq 0$, so $x \geq 4$

Applications of Function Composition

In Calculus

Function composition is essential for the chain rule in differentiation: If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.

In Real-World Problems

Function composition models sequential processes. For example:

• Temperature conversion: Convert Fahrenheit to Kelvin by first converting F to C, then C to K
• Business: Apply a discount to a price, then add sales tax
• Physics: Velocity is the derivative of position, acceleration is the derivative of velocity

Examples

Example 1: Polynomial Functions

Let $f(x) = 2x + 3$ and $g(x) = x^2 - 1$. Find $(f \circ g)(x)$.

Solution:

1. $(f \circ g)(x) = f(g(x))$
2. Substitute $g(x) = x^2 - 1$ into $f(x) = 2x + 3$:
3. $f(x^2 - 1) = 2(x^2 - 1) + 3$
4. $= 2x^2 - 2 + 3$
5. $= 2x^2 + 1$

Example 2: Rational and Polynomial Functions

Let $f(x) = \frac{1}{x}$ and $g(x) = x + 2$. Find both $(f \circ g)(x)$ and $(g \circ f)(x)$.

Solution:

1. $(f \circ g)(x) = f(g(x)) = f(x + 2) = \frac{1}{x + 2}$
2. $(g \circ f)(x) = g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{x} + 2 = \frac{1 + 2x}{x}$
3. Notice: $(f \circ g)(x) \neq (g \circ f)(x)$

Example 3: Verifying Inverse Functions

Let $f(x) = 2x + 3$ and $g(x) = \frac{x - 3}{2}$. Verify that f and g are inverses.

Solution:

1. Check $(f \circ g)(x)$: $f\left(\frac{x - 3}{2}\right) = 2 \cdot \frac{x - 3}{2} + 3 = x - 3 + 3 = x$ ✓
2. Check $(g \circ f)(x)$: $g(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x$ ✓
3. Since both compositions equal x, f and g are inverses.

Tips for Using This Calculator

• Enter functions using x as the variable
• Use * for multiplication (e.g., 2*x instead of 2x)
• Use ^ or ** for exponents (e.g., x^2 or x**2)
• Use sqrt(x) for square root
• Use log(x) for natural logarithm
• Use exp(x) or e^x for exponential function
• Use parentheses to clarify order of operations

Frequently Asked Questions

What is the difference between (f ∘ g)(x) and f(x) × g(x)?

$(f \circ g)(x)$ is function composition, meaning $f(g(x))$. In contrast, $f(x) \times g(x)$ is function multiplication, where you multiply the outputs of both functions. These are completely different operations.

How do I read the notation (f ∘ g)(x)?

Read it as "f composed with g of x" or simply "f of g of x". The small circle ∘ indicates composition, not multiplication.

Does order matter in function composition?

Yes! Function composition is not commutative. $(f \circ g)(x)$ usually gives a different result than $(g \circ f)(x)$. Always pay attention to which function is applied first.

How do I find the domain of a composite function?

The domain of $(f \circ g)(x)$ consists of all x-values where: (1) x is in the domain of g, AND (2) $g(x)$ is in the domain of f. You must check both conditions.

Additional Resources

To learn more about function composition:

• Function Composition - Wikipedia
• Function Composition - Khan Academy
• Function Composition - Wolfram MathWorld

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