Problem 12
Question
Use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. $$ f(x)=x^{2}+1, g(x)=\sqrt{x+2} $$
Step-by-Step Solution
Verified Answer
\( f(g(x)) = x + 3 \); \( g(f(x)) = \sqrt{x^2 + 3} \).
1Step 1: Understand the Composition of Functions
The composition of two functions, such as \( f(g(x)) \), means that we plug the function \( g(x) \) into the function \( f(x) \). Similarly, for \( g(f(x)) \), we plug \( f(x) \) into \( g(x) \). This process involves substituting the second function into the first and then simplifying.
2Step 2: Calculate \( f(g(x)) \)
First, write down the expression for \( g(x) = \sqrt{x+2} \). We will substitute this into \( f(x) = x^2 + 1 \). So, \( f(g(x)) = f(\sqrt{x+2}) = (\sqrt{x+2})^2 + 1 \). Simplify this: \( (\sqrt{x+2})^2 = x+2 \). Therefore, \( f(g(x)) = x + 2 + 1 = x + 3 \).
3Step 3: Calculate \( g(f(x)) \)
Next, write down the expression for \( f(x) = x^2 + 1 \). We will substitute this into \( g(x) = \sqrt{x+2} \). So, \( g(f(x)) = g(x^2 + 1) = \sqrt{x^2 + 1 + 2} \). Simplify this: \( \sqrt{x^2 + 3} \). Thus, \( g(f(x)) = \sqrt{x^2 + 3} \).
4Step 4: Simplify and Summarize
From the calculations in Steps 2 and 3, we have found that \( f(g(x)) = x + 3 \) and \( g(f(x)) = \sqrt{x^2 + 3} \). Both expressions are fully simplified, with no remaining calculations required.
Key Concepts
Composite FunctionsSimplificationSubstitution in Functions
Composite Functions
Understanding composite functions might seem like a puzzle at first, but it's a clever way to blend two functions into one. When you see something like \( f(g(x)) \), remember that we are essentially nesting functions. This means taking the output from the function \( g(x) \) and feeding it directly into \( f(x) \).
For example, with the given functions, \( f(x) = x^2 + 1 \) and \( g(x) = \sqrt{x+2} \), we compose \( f(g(x)) \) to get \((\sqrt{x+2})^2 + 1\).
This shows how composite functions interplay by transforming one input through multiple stages.
For example, with the given functions, \( f(x) = x^2 + 1 \) and \( g(x) = \sqrt{x+2} \), we compose \( f(g(x)) \) to get \((\sqrt{x+2})^2 + 1\).
This shows how composite functions interplay by transforming one input through multiple stages.
Simplification
Simplification is like organizing your thoughts on paper — making things as straightforward as possible. After composing a function, it's essential to simplify it. This means combining like terms, getting rid of unnecessary parts, and making the expression easy to understand.
When we computed \( f(g(x)) \) and simplified it to \( x + 3 \), we squared \( \sqrt{x+2} \) into \( x + 2 \), and then simply added 1.
Similarly, for \( g(f(x)) \), we aimed for \( \sqrt{x^2+3} \), removing any extra steps to make the expression clear and concise.
When we computed \( f(g(x)) \) and simplified it to \( x + 3 \), we squared \( \sqrt{x+2} \) into \( x + 2 \), and then simply added 1.
Similarly, for \( g(f(x)) \), we aimed for \( \sqrt{x^2+3} \), removing any extra steps to make the expression clear and concise.
Substitution in Functions
Substitution in functions is like filling in blanks in a story. You take an expression or another function and place it into the variable slot of the primary function. This crucial step lets you explore new relationships between mathematical expressions.
In the original problem, we substituted \( g(x) = \sqrt{x+2} \) into \( f(x) \), effectively replacing every \( x \) in \( f(x) \) with \( g(x) \).
Reversing this process, we placed \( f(x) = x^2 + 1 \) into \( g(x) \), showing flexibility and interchangeability in functions.
In the original problem, we substituted \( g(x) = \sqrt{x+2} \) into \( f(x) \), effectively replacing every \( x \) in \( f(x) \) with \( g(x) \).
Reversing this process, we placed \( f(x) = x^2 + 1 \) into \( g(x) \), showing flexibility and interchangeability in functions.
Other exercises in this chapter
Problem 12
For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. $$ f(x)=-5|x+2|+15 $$
View solution Problem 12
Describe how the graph of the function is a transformation of the graph of the original function \(f.\) $$y=f(x+3)$$
View solution Problem 12
For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x)) .\) Simplify your answers. $$f(x)=x^{2}+1, g(x)=\sqrt{x+2}$$
View solution Problem 12
For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. $$ b(x
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