5.3 Composition of Functions

Function composition is a powerful tool in mathematics that combines two or more functions into a single operation. It's like a mathematical assembly line, where the output of one function becomes the input for another. This concept is crucial for modeling complex systems and solving real-world problems.

In this section, we'll explore how to compose functions, determine their domains and ranges, and apply these skills to practical situations. We'll see how function composition can simplify calculations and provide insights into multi-step processes across various fields.

Composition of Functions

Definition and Properties

Top images from around the web for Definition and Properties

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Function Composition View original

  Is this image relevant?

• 

  Composition of Functions - The Bearded Math Man View original

  Is this image relevant?

• 

  Compositions of Functions | College Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Function Composition View original

  Is this image relevant?

• 

  Composition of Functions - The Bearded Math Man View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Properties

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Function Composition View original

  Is this image relevant?

• 

  Composition of Functions - The Bearded Math Man View original

  Is this image relevant?

• 

  Compositions of Functions | College Algebra View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Function Composition View original

  Is this image relevant?

• 

  Composition of Functions - The Bearded Math Man View original

  Is this image relevant?

1 of 3

• The composition of two functions $f$ and $g$, denoted $(f \circ g)(x)$, is defined as $[f(g(x))](https://www.fiveableKeyTerm:f(g(x)))$, where the output of function $g$ becomes the input of function $f$
• For the composition $(f \circ g)(x)$ to be defined, the range of $g$ must be a subset of the domain of $f$
  • This ensures that the output values of $g$ are valid inputs for $f$
  • If this condition is not met, the composition may not be well-defined for all values in the domain of $g$
• Function composition is not commutative, meaning $(f \circ g)(x)$ is not always equal to $(g \circ f)(x)$
  • The order in which functions are composed matters
  • For example, let $f(x) = x^2$ and $g(x) = x + 1$. Then, $(f \circ g)(x) = (x + 1)^2$, while $(g \circ f)(x) = x^2 + 1$
• Function composition is associative, meaning $((f \circ g) \circ h)(x) = (f \circ (g \circ h))(x)$
  • This property allows for the grouping of functions in a composition without affecting the result
  • For example, let $f(x) = 2x$, $g(x) = x + 1$, and $h(x) = x^2$. Then, $((f \circ g) \circ h)(x) = (f \circ (g \circ h))(x) = 2((x^2 + 1))$
• The identity function, denoted $I(x) = x$, has the property that $(f \circ I)(x) = (I \circ f)(x) = f(x)$ for any function $f$
  • Composing a function with the identity function does not change the original function
  • For example, if $f(x) = 3x - 1$, then $(f \circ I)(x) = (I \circ f)(x) = 3x - 1$

Notation and Terminology

• The symbol $\circ$ is used to denote function composition, as in $(f \circ g)(x)$
• The function on the right side of the composition symbol is applied first, followed by the function on the left side
  • In $(f \circ g)(x)$, $g$ is applied to $x$ first, and then $f$ is applied to the result
• The function $g$ in the composition $(f \circ g)(x)$ is called the "inner function" or "inside function"
• The function $f$ in the composition $(f \circ g)(x)$ is called the "outer function" or "outside function"

Computing Function Compositions

Evaluating Composite Functions

• To find the composition $(f \circ g)(x)$, first evaluate $g(x)$, then use the result as the input for $f$
  • For example, let $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. To find $(f \circ g)(4)$:
    1. Evaluate $g(4) = 2(4) - 3 = 5$
    2. Use the result as the input for $f$: $f(5) = 5^2 + 1 = 26$
    3. Therefore, $(f \circ g)(4) = 26$
• When composing more than two functions, work from the innermost function outward, following the order of operations
  • For example, let $f(x) = \sqrt{x}$, $g(x) = x^2 + 1$, and $h(x) = 3x - 2$. To find $(f \circ g \circ h)(x)$:
    1. Start with the innermost function, $h(x) = 3x - 2$
    2. Apply $g$ to the result: $g(3x - 2) = (3x - 2)^2 + 1 = 9x^2 - 12x + 5$
    3. Apply $f$ to the result: $f(9x^2 - 12x + 5) = \sqrt{9x^2 - 12x + 5}$
    4. Therefore, $(f \circ g \circ h)(x) = \sqrt{9x^2 - 12x + 5}$
• Simplify the resulting expression by combining like terms and applying any necessary algebraic manipulations
  • For example, let $f(x) = 2x + 1$ and $g(x) = 3x - 4$. To find $(f \circ g)(x)$:
    1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = 2(3x - 4) + 1$
    2. Simplify the expression: $2(3x - 4) + 1 = 6x - 8 + 1 = 6x - 7$
    3. Therefore, $(f \circ g)(x) = 6x - 7$
• Be cautious when composing functions with different domains, as the resulting composite function may have a more restricted domain
  • For example, let $f(x) = \sqrt{x}$ and $g(x) = x - 4$. The domain of $f$ is $[0, \infty)$, while the domain of $g$ is $(-\infty, \infty)$. The domain of $(f \circ g)(x)$ is $[4, \infty)$ because $g(x)$ must be non-negative for $f(g(x))$ to be defined.

Domain and Range of Composite Functions

Determining the Domain

• The domain of $(f \circ g)(x)$ consists of all $x$ values in the domain of $g$ for which $g(x)$ is in the domain of $f$
  • In other words, the domain of the composite function is the set of all inputs for which both $g$ and $f$ are defined
• To find the domain of a composite function, first determine the domain of the innermost function, then consider any restrictions imposed by the outer function(s)
  • For example, let $f(x) = \sqrt{x}$ and $g(x) = x^2 - 4$. The domain of $g$ is $(-\infty, \infty)$, but the domain of $f$ is $[0, \infty)$. For $(f \circ g)(x)$ to be defined, $g(x)$ must be non-negative:
    1. Set up the inequality: $x^2 - 4 \geq 0$
    2. Solve the inequality: $(x - 2)(x + 2) \geq 0$, which gives $x \leq -2$ or $x \geq 2$
    3. Therefore, the domain of $(f \circ g)(x)$ is $(-\infty, -2] \cup [2, \infty)$
• When composing multiple functions, consider the restrictions imposed by each function in the composition
  • For example, let $f(x) = \ln(x)$, $g(x) = \sqrt{x}$, and $h(x) = x^2 - 1$. The domain of $h$ is $(-\infty, \infty)$, the domain of $g$ is $[0, \infty)$, and the domain of $f$ is $(0, \infty)$. For $(f \circ g \circ h)(x)$ to be defined:
    1. $h(x)$ must be non-negative for $g$ to be defined: $x^2 - 1 \geq 0$, which gives $x \leq -1$ or $x \geq 1$
    2. $g(h(x))$ must be positive for $f$ to be defined: $\sqrt{x^2 - 1} > 0$, which gives $x < -1$ or $x > 1$
    3. Combining the inequalities, the domain of $(f \circ g \circ h)(x)$ is $(-\infty, -1) \cup (1, \infty)$

Finding the Range

• The range of $(f \circ g)(x)$ is the set of all possible output values when the composite function is evaluated over its entire domain
  • In other words, the range is the set of all values that $(f \circ g)(x)$ can produce given its domain
• To find the range of a composite function, consider the range of the outer function and any restrictions imposed by the inner function(s)
  • For example, let $f(x) = x^2$ and $g(x) = x + 1$. The range of $g$ is $(-\infty, \infty)$, and the range of $f$ is $[0, \infty)$. To find the range of $(f \circ g)(x)$:
    1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = (x + 1)^2 = x^2 + 2x + 1$
    2. The minimum value of $(f \circ g)(x)$ occurs at $x = -1$, which gives $(f \circ g)(-1) = 0$
    3. As $x$ approaches $\infty$ or $-\infty$, $(f \circ g)(x)$ approaches $\infty$
    4. Therefore, the range of $(f \circ g)(x)$ is $[0, \infty)$
• In some cases, finding the range of a composite function may require more advanced techniques, such as calculus or graphical analysis
  • For example, let $f(x) = \sin(x)$ and $g(x) = x^2$. The range of $g$ is $[0, \infty)$, and the range of $f$ is $[-1, 1]$. To find the range of $(f \circ g)(x)$:
    1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = \sin(x^2)$
    2. As $x$ approaches $\infty$ or $-\infty$, $\sin(x^2)$ oscillates between $-1$ and $1$
    3. Therefore, the range of $(f \circ g)(x)$ is $[-1, 1]$, which is the same as the range of $f$

Function Composition for Real-World Problems

Modeling Multi-Step Processes

• Function composition can be used to model multi-step processes or systems where the output of one function serves as the input for another
  • For example, in a manufacturing process, the output of a raw material processing function may serve as the input for a component assembly function, which in turn provides input for a final product packaging function
• Real-world examples of function composition include:
  • Compound interest: The principal amount (initial investment) is first increased by the interest rate, and then this new amount becomes the principal for the next compounding period
  • Population growth: The initial population is affected by factors such as birth rate, death rate, and migration, and the resulting population becomes the starting point for the next time period
  • Supply chain management: The output of a supplier (raw materials) becomes the input for a manufacturer, whose output (finished products) becomes the input for a distributor, and so on until the product reaches the end customer
• When modeling real-world problems using function composition, it is essential to clearly define the individual functions and their domains
  • This ensures that the composed functions accurately represent the relationships between the various components of the system being modeled
  • For example, in a population growth model, the birth rate function may have a domain of $[0, 1]$ (representing the percentage of the population that gives birth each year), while the death rate function may have a similar domain but with different values

Solving Problems and Interpreting Results

• When solving problems using function composition, follow these steps:
  1. Identify the individual functions that make up the multi-step process or system
  2. Determine the domains and ranges of each function
  3. Compose the functions in the appropriate order
  4. Simplify the resulting composite function, if possible
  5. Evaluate the composite function for specific input values or analyze its behavior over its entire domain
• Interpret the results of the composite function in the context of the problem, considering any limitations or assumptions made in the modeling process
  • For example, in a compound interest problem, the output of the composite function represents the account balance after a certain number of compounding periods
  • It is important to consider factors such as the initial investment amount, interest rate, and compounding frequency when interpreting the results
• Verify that the solution makes sense given the constraints of the real-world situation being modeled
  • This may involve comparing the results to historical data, consulting with subject matter experts, or conducting sensitivity analyses to assess the impact of changes in input values or assumptions
  • For example, in a population growth model, the projected population should not exceed the carrying capacity of the environment (the maximum population that can be sustained given available resources)
• When communicating the results of a problem solved using function composition, be sure to:
  • Clearly state the assumptions and limitations of the model
  • Provide context for the input values and resulting outputs
  • Use visualizations (graphs, charts, tables) to help convey the relationships between the components of the system and the behavior of the composite function
  • Discuss any potential implications or recommendations based on the insights gained from the analysis