Problem 83
Question
For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of \((f \circ g)(x) ?\)
Step-by-Step Solution
Verified Answer
The domain of \((f \circ g)(x)\) is all real numbers.
1Step 1: Understand the composition function
We are asked to find the domain of \((f \circ g)(x)\), which means \((f \circ g)(x) = f(g(x))\). In this case, \(f(x)=x^3+1\) and \(g(x)=\sqrt[3]{x-1}\), so \(f(g(x)) = ((\sqrt[3]{x-1})^3 + 1)\).
2Step 2: Simplify the composition
Substituting \(g(x)\) into \(f(x)\), we have \(((\sqrt[3]{x-1})^3 + 1)\). The cube root and cube are inverse functions, so we have \(f(g(x)) = (x - 1) + 1 = x\). Thus \((f \circ g)(x) = x\).
3Step 3: Determine the domain of the composition
The function \((f \circ g)(x) = x\) is a linear function and it is defined for all real numbers. Hence, the domain is all real numbers.
Key Concepts
Domain of a FunctionLinear FunctionInverse Functions
Domain of a Function
The domain of a function is a crucial concept in mathematics. It refers to the set of all possible input values for which the function is defined. In simpler terms, it's the set of values you can safely plug into the function without causing any mathematical mishaps, like division by zero or taking the square root of a negative number.
When dealing with composed functions, such as \(f \circ g\), determining the domain requires understanding both functions involved. With the exercise at hand, we have \(f(x) = x^3 + 1\) and \(g(x) = \sqrt[3]{x-1}\).
The function composition \(f \circ g(x) = f(g(x))\) simplifies to \(f(g(x)) = ((\sqrt[3]{x-1})^3 + 1)\). The cube root and cube cancel each other out, resulting in \(f(g(x)) = x\). Since \(x\) is a simple linear function, it is defined for all real numbers. Thus, the domain of \(f \circ g(x)\) is all real numbers because there are no restrictions on \(x\) that could lead to undefined expressions.
To summarize:
When dealing with composed functions, such as \(f \circ g\), determining the domain requires understanding both functions involved. With the exercise at hand, we have \(f(x) = x^3 + 1\) and \(g(x) = \sqrt[3]{x-1}\).
The function composition \(f \circ g(x) = f(g(x))\) simplifies to \(f(g(x)) = ((\sqrt[3]{x-1})^3 + 1)\). The cube root and cube cancel each other out, resulting in \(f(g(x)) = x\). Since \(x\) is a simple linear function, it is defined for all real numbers. Thus, the domain of \(f \circ g(x)\) is all real numbers because there are no restrictions on \(x\) that could lead to undefined expressions.
To summarize:
- Domain Definition: All values for which a function is defined.
- Composition: Consider each function in the composition when determining the domain.
- Result: The function \(f \circ g(x) = x\) is defined for all real numbers.
Linear Function
Linear functions are one of the simplest types of functions you will encounter in mathematics. They have the general form of \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.
In the exercise, after composing \(f(x)\) and \(g(x)\), we found that the function simplifies to \(f \circ g(x) = x\), which is a linear function with a slope of 1 and no y-intercept (often expressed as \(y = x + 0\)).
Linear functions are defined for all real numbers without any restrictions. This is due to the fact that, in a linear function, each input corresponds to exactly one output, creating a straight line when graphed.
Key points about linear functions:
In the exercise, after composing \(f(x)\) and \(g(x)\), we found that the function simplifies to \(f \circ g(x) = x\), which is a linear function with a slope of 1 and no y-intercept (often expressed as \(y = x + 0\)).
Linear functions are defined for all real numbers without any restrictions. This is due to the fact that, in a linear function, each input corresponds to exactly one output, creating a straight line when graphed.
Key points about linear functions:
- Form: \(y = mx + b\)
- Graph: Always a straight line.
- Domain: All real numbers (\(-\infty, \infty\)).
Inverse Functions
Inverse functions are pivotal when it comes to understanding how two functions can "undo" each other. Think of them as mathematical operations that are opposite in nature. If you perform a function and then its inverse, you end up back where you started.
In this exercise, we observed this property when dealing with the functions \(f(x) = x^3 + 1\) and \(g(x) = \sqrt[3]{x-1}\). When you compose these functions, the cube root \(\sqrt[3]{x-1}\) and the cube operation \(x^3\) cancel each other.
Here's how inverse functions work:
In this exercise, we observed this property when dealing with the functions \(f(x) = x^3 + 1\) and \(g(x) = \sqrt[3]{x-1}\). When you compose these functions, the cube root \(\sqrt[3]{x-1}\) and the cube operation \(x^3\) cancel each other.
Here's how inverse functions work:
- Definition: If \(f(a) = b\) then the inverse, denoted as \(f^{-1}(b) = a\).
- In our exercise, \(g(x)\) is the inverse of \(f(x)\) when corrected for the constant shift, as they cancel each other's effects on the variable \(x\).
- Result: This cancellation leads to the outcome \(f \circ g(x) = x\), essentially getting back the original input.
Other exercises in this chapter
Problem 82
For the following exercises, use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). What is the domain of $$ (g \circ f)(x) ? $$
View solution Problem 82
For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. \([0,
View solution Problem 83
For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. \([0,
View solution Problem 84
Let \(f(x)=\frac{1}{x}\). (a) Find \((f \circ f)(x)\). (b) Is \((f \circ f)(x)\) for any function \(f\) the same result as the answer to part (a) for any functi
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