Composition of Functions: Videos & Practice Problems

Function composition involves substituting one function into another, such as $f_g\left(x\right)=f_{\left(}$. Evaluating composed functions can be done by first composing and then substituting values or by sequentially evaluating inner and outer functions. Finding the domain requires considering restrictions from both functions, such as avoiding negative values under square roots or zero denominators. Simplifying expressions often uses FOIL for binomials, ensuring results are in standard form with descending powers. Understanding these concepts is essential for mastering polynomial and function operations.

Function Compositions

Function composition is a fundamental concept in mathematics that involves combining two functions to create a new function. When composing functions, instead of substituting a number into a function, you substitute one function into another. This process can initially seem challenging, but with practice, it becomes clearer.

To evaluate a function at a specific value, you replace the variable with that value. For example, if you have the function \( f(x) = x^2 + 3x - 10 \) and you want to evaluate it at \( f(7) \), you would calculate:

\( f(7) = 7^2 + 3(7) - 10 = 49 + 21 - 10 = 60 \).

In contrast, when composing functions, such as finding \( f(g(x)) \), you replace the variable in \( f(x) \) with the entire function \( g(x) \). For instance, if \( g(x) = x - 2 \), then:

\( f(g(x)) = f(x - 2) = (x - 2)^2 + 3(x - 2) - 10 \).

Upon simplifying, this results in:

\( f(g(x)) = x^2 - x - 12 \).

Function composition can be denoted as \( f \circ g \), which indicates that \( g \) is applied first, followed by \( f \). The notation emphasizes that \( f \) is the outer function and \( g \) is the inner function.

To illustrate further, consider the functions \( f(x) = x + 4 \) and \( g(x) = x^2 - 3 \). To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):

\( f(g(x)) = f(x^2 - 3) = (x^2 - 3) + 4 = x^2 + 1 \).

For the reverse composition \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \):

\( g(f(x)) = g(x + 4) = (x + 4)^2 - 3 \).

Expanding this using the FOIL method gives:

\( g(f(x)) = x^2 + 8x + 16 - 3 = x^2 + 8x + 13 \).

In summary, function composition allows you to create new functions by substituting one function into another, and understanding this process is crucial for solving more complex mathematical problems.

Given the functions $f\left(x\right)=\sqrt{x+4}$ and $g\left(x\right)={(x-2)}^2-4$, find $(f\circ g)\left(x\right)$ and $(g\circ f)\left(x\right)$.

• A. $\left(f\circ g\right)\left(x\right)=\sqrt{x-2}$ ; $\left(g\circ f\right)\left(x\right)=\left(x+4\right)-4\sqrt{x+4}$
• B. $\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=x\left(x+4\right)$
• C. $\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=4\sqrt{x-4}$
• D. $\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=\left(x+4\right)-4\sqrt{x+4}$

Answer: (D)

Given the functions $f\left(x\right)=\frac{1}{x^2-2}$ and $g\left(x\right)=\sqrt{x+2}$, find $(f\circ g)\left(x\right)$ and $(g\circ f)\left(x\right)$

• A. $(f\circ g)(x)=\frac{1}{x}$ ; $(g\circ f)(x)=\sqrt{\frac{2x^2-3}{x^2-2}}$
• B. $(f\circ g)(x)=\frac{1}{x}$ ; $(g\circ f)(x)=\sqrt{\frac{3}{x^2-2}}$
• C. $(f∘g)(x)=x$ ; $(g\circ f)(x)=\sqrt{x^2-2}$
• D. $(f∘g)(x)=x$ ; $(g\circ f)(x)=\frac{1}{\sqrt{x+2}-2}$

Answer: (A)

Evaluating Composed Functions

Evaluating composed functions involves understanding how to combine two functions and then determine their output based on a specific input. When given two functions, such as f(x) = x² and g(x) = x - 1, the composition f(g(x)) can be found by substituting g(x) into f(x). This process begins by calculating f(g(x)) = f(x - 1), which simplifies to (x - 1)². By applying the FOIL method, we expand this to x² - 2x + 1, providing the composed function.

To evaluate this composition at a specific value, such as 3, we replace x in the composed function with 3. This results in f(g(3)) = 3² - 2(3) + 1, which simplifies to 9 - 6 + 1 = 4. Thus, f(g(3)) = 4.

Alternatively, a shortcut method can be employed for evaluating composed functions. This involves first calculating the inner function at the desired input. For instance, finding g(3) gives us 3 - 1 = 2. We then substitute this result into the outer function: f(g(3)) = f(2), which simplifies to 2² = 4. This method can be quicker but is not always applicable, especially when the problem requires finding the composition before evaluating.

In summary, while both methods yield the same result, the choice of method may depend on the specific requirements of the problem. Understanding how to compose functions and evaluate them effectively is crucial in mastering function operations.

Given the functions $f\left(x\right)=x+3$ and $g\left(x\right)=x^2$, find $(f\circ g)\left(2\right)$ and $(g\circ f)\left(2\right)$.

• A. $(f∘g)(2)=5$ ; $(g∘f)(2)=25$
• B. $(f\circ g)(2)=7;(g\circ f)(2)=4$
• C. $(f∘g)(2)=7$ ; $(g∘f)(2)=25$
• D. $(f∘g)(2)=1$ ; $(g∘f)(2)=1$

Answer: (C)

Domain Restrictions of Composed Functions

Understanding the domain of composed functions is essential for evaluating them correctly. When dealing with composed functions, such as \( f(g(x)) \), it is crucial to identify the restrictions imposed by both the inner function \( g(x) \) and the outer function \( f(x) \).

Consider the functions \( f(x) = \frac{1}{x - 2} \) and \( g(x) = \sqrt{x} \). To find the domain of the composition \( f(g(x)) \), we follow a systematic approach. First, we determine the restrictions for the inner function \( g(x) \). Since \( g(x) = \sqrt{x} \), it is clear that \( x \) must be greater than or equal to 0, as the square root of a negative number is undefined. Thus, the restriction for \( g(x) \) is:

\( x \geq 0 \)

Next, we substitute \( g(x) \) into \( f(x) \) to form the composed function:

\( f(g(x)) = f(\sqrt{x}) = \frac{1}{\sqrt{x} - 2} \)

Now, we need to identify any additional restrictions that arise from this composition. Specifically, the denominator of the fraction cannot equal zero. Therefore, we set up the equation:

\( \sqrt{x} - 2 \neq 0 \)

Solving this gives:

\( \sqrt{x} \neq 2 \)

Squaring both sides results in:

\( x \neq 4 \)

At this point, we have two restrictions: \( x \geq 0 \) from \( g(x) \) and \( x \neq 4 \) from the composition. To express the domain of the composed function \( f(g(x)) \), we combine these restrictions. The domain can be described as all values from 0 to 4, excluding 4, and all values greater than 4:

Domain: \( [0, 4) \cup (4, \infty) \)

This comprehensive approach ensures that we account for all restrictions when determining the domain of composed functions, allowing for accurate evaluations and further mathematical operations.

Given the functions $f\left(x\right)=x^2$ and $g\left(x\right)=\sqrt{x-8}$, find $(f\circ g)\left(x\right)$ and determine its domain.

• A. $(f∘g)(x)=x-8$ ; $Dom:(-\infty,\infty)$
• B. $(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:(-∞,∞)$
• C. $(f∘g)(x)=x-8$ ; $Dom:[8,∞)$
• D. $(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:[8,∞)$

Answer: (C)

Here’s what students ask on this topic:

What is the composition of functions and how do you compute $f_g\left(x\right)$?

Function composition involves substituting one function into another. To compute $f_{\circ }\left(x\right)$ (also written as $f_g\left(x\right)$), you replace the variable $x$ in the outer function $f$ with the entire inner function $g\left(x\right)$. For example, if $f\left(x\right)=x^2+\; 3x-\; 10$ and $g\left(x\right)=x-\; 2$, then $f_g\left(x\right)={(<mi>x</mi>\; -\; 2)}^2+\; 3(x-\; 2)\; -\; 10$. Simplify this expression by expanding and combining like terms to get the composed function.

How do you evaluate a composed function like $f_g\left(3\right)$?

To evaluate a composed function such as $f_g\left(3\right)$, you can use two methods. First, compose the functions to get $f_g\left(x\right)$ and then substitute $3$ for $x$. Alternatively, use the shortcut: first evaluate the inner function at 3, i.e., find $g\left(3\right)$, then plug that result into the outer function, i.e., find $f\left(g\right(3)$. For example, if $f\left(x\right)=x^2$ and $g\left(x\right)=x-\; 1$, then $g\left(3\right)=\; 2$ and $f\left(2\right)=\; 4$. This method is often faster but only works when you are asked to evaluate at a number directly.

How do you find the domain of a composed function like $f_g\left(x\right)$?

To find the domain of a composed function $f_g\left(x\right)$, first determine the domain of the inner function $g\left(x\right)$. Then find the domain restrictions of the outer function $f$ when its input is $g\left(x\right)$. For example, if $f\left(x\right)=\; 1/(x-\; 2)$ and $g\left(x\right)=\sqrt{x}$, the domain of $g$ is $x2\; 2\; 0$ because the radicand must be non-negative. Next, ensure the denominator in $f\left(g\right(x)$ is not zero, so $\sqrt{x}-\; 2\; 2\; 2\; 0$ or $x2\; 2\; 4$. Combining these, the domain is $x2\; 2\; 0$ and $x2\; 2\; 4$ but excluding $x=\; 4$. This process ensures the composed function is defined.

What is the difference between $f_g\left(x\right)$ and $g_f\left(x\right)$?

The difference lies in the order of composition. $f_g\left(x\right)$ means you apply $g$ first, then apply $f$ to the result of $g$. Conversely, $g_f\left(x\right)$ means you apply $f$ first, then apply $g$ to the result of $f$. These two compositions generally produce different functions. For example, if $f\left(x\right)=x+\; 4$ and $g\left(x\right)=x^2-\; 3$, then $f_g\left(x\right)=\; (x^2-\; 3)\; +\; 4\; =x^2+\; 1$, but $g_f\left(x\right)=\; (x+\; 4)2-\; 3\; =x^2+\; 8x+\; 13$. The order affects the result.

How do you simplify the expression after composing two functions?

After composing two functions, you often get an expression that requires simplification. This usually involves expanding binomials using the FOIL method (First, Outer, Inner, Last) and combining like terms. For example, if you have $(x+\; 4)2$, expand it as $(x+\; 4)(x+\; 4)$. Multiply: First terms $ximesx=x^2$, Outer terms $ximes\; 4\; =\; 4x$, Inner terms $4\; imesx=\; 4x$, Last terms $4\; imes\; 4\; =\; 16$. Combine like terms: $4x+\; 4x=\; 8x$. The simplified expression is $x^2+\; 8x+\; 16$. Simplification helps in understanding the composed function's behavior and makes further calculations easier.