Composite Functions
Composite Functions are functions that result from applying one function to the output of another function. They can be expressed algebraically as:
In simpler terms, if you have two functions \( f(x) \) and \( g(x) \), then the composite function \( (f \circ g)(x) \) is defined as \( f(g(x)) \). Similarly, the composite function \( (g \circ f)(x) \) is defined as \( g(f(x)) \).
It's important to note that the results of the composite function depends on the inner function and the outer function. Other characteristics, such as the domain and range, will also vary. Unless \(f(x)\) and \(g(x)\) are the exact same,
To evaluate a composite function, \(f(g(x))\), at a specific value, substitute the value into the composite function and simplify.
Alternatively, you can evaluate \(g(x)\), at the specific value and substitute the value into \(f(x)\).
Example
If \(f(x) = 2x\) and \(g(x) = x + 3\), determine the following composite functions. After, sketch a graph of the function and determine its domain and range.
- \( (f \circ g)(x) \)
- \( (g \circ f)(x) \)
i. To determine the composite function \( (f \circ g)(x) \), we can substitute \(g(x)\) for \(x\) in \(f(x)\):
\(f(x) = 2x\)
\((f \circ g)(x) = 2(g(x))\)
Next, we can expand the function to get our final answer:
\((f \circ g)(x) = 2(x+ 3)\)
\((f \circ g)(x) = 2x + 6\)
Therefore, we can determine that \(\boldsymbol{(f \circ g)(x) = 2x + 6}\).
We can sketch a graph of the function as such:
From this graph, we can determine the domain is \(\boldsymbol{\{x\in\mathbb{R}\}}\) and the range is \(\boldsymbol{\{y\in\mathbb{R}\}}\).
ii. To determine the composite function \( (g \circ f)(x) \), we can substitute \(f(x)\) for \(x\) in \(g(x)\):
\(g(x) = x + 3 \)
\((g \circ f)(x) = g(x) + 3\)
Next, we can expand the function to get our final answer:
\((f \circ g)(x) = 2x + 3\)
Therefore, we can determine that \(\boldsymbol{(g \circ f)(x) = 2x + 3}\).
We can sketch a graph of the function as such:
From this graph, we can determine the domain is \(\boldsymbol{\{x\in\mathbb{R}\}}\) and the range is \(\boldsymbol{\{y \in \mathbb{R}\}}\).
\((f \circ g)(x)\)
To determine the composite function \( (f \circ g)(x) \), we can substitute \(g(x)\) for \(x\) in \(f(x)\):
\(f(x) = x^2\)
\((f \circ g)(x) = (g(x))^2\)
Next, we can expand the function to get our final answer:
\((f \circ g)(x) = (x-4)^2\)
\((f \circ g)(x) = x^2 - 8x + 16\)
Therefore, we can determine that \(\boldsymbol{(f \circ g)(x) = x^2 - 8x + 16}\).
\((g \circ f)(x)\)
To determine the composite function \((g \circ f)(x) \), we can substitute \(g(x)\) for \(x\) in \(f(x)\):
\(g(x) = x - 4\)
\((g \circ f)(x) = g(x) - 4\)
Next, we can expand the function to get our final answer:
\((g \circ f)(x) = x^2 - 4\)
Therefore, we can determine that \(\boldsymbol{(g \circ f)(x) = x^2 - 4}\).
\((f \circ f)(x)\)
To determine the composite function \((f \circ f)(x) \), we can substitute \(f(x)\) for \(x\) in \(f(x)\):
\(f(x) = x - 4\)
\((f \circ f)(x) = f(x) - 4\)
Next, we can expand the function to get our final answer:
\((f \circ f)(x) = x-4 - 4\)
\((f \circ f)(x) = x-8\)
Therefore, we can determine that \(\boldsymbol{(f \circ f)(x) = x^2 - 4}\).
\((g \circ g)(x)\)
To determine the composite function \((g \circ g)(x) \), we can substitute \(g(x)\) for \(x\) in \(g(x)\):
\(g(x) = x^2\)
\((g \circ g)(x) = (g(x))^2\)
Next, we can expand the function to get our final answer:
\((g \circ g)(x) = (x^2)^2\)
\((g \circ g)(x) = x^4\)
Therefore, we can determine that \(\boldsymbol{(g \circ g)(x) = x^4}\).
Composite Function Calculator
Select the coefficient and term values for \(f(x)\) and \(g(x)\). This function will then determine the composite functions for \((f \cdot g)(x)\) and \((g \cdot f)(x)\).