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Chapter 2: Problem 74

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=x^{2}+1, g(x)=\sqrt{2-x}$$

Short Answer

Expert verified
\((f \circ g)(x) = 3 - x\) and the domain of \(f \circ g\) is \(x \leq 2\).

Step by step solution

01

Find the Composition of Two Functions

First, let's find the composition of \(f\) and \(g\), which is denoted by \((f \circ g)(x)\). It simply means that for a given \(x\), replace \(f(x)\) with \(g(x)\). So, we have \(f(g(x))\).We know, \(f(x) = x^{2}+1\) and \(g(x) = \sqrt{2-x}\). Thus, replacing \(x\) in function \(f(x)\) with \(g\), we get: \(f(g(x)) = (\sqrt{2-x})^{2}+1 = 2 - x + 1 = 3 - x\) .
02

Find the Domain of \(f \circ g\)

Now, in the second step, let's find the domain of \(f \circ g\). The domain of \(f(g(x))\) is the set of all real numbers \(x\) that make \(g(x)\) valid and also, these numbers should be in the domain of \(f\). From \(g(x) = \sqrt{2-x}\), we can see that \(2-x\) should be greater than or equal to 0 to make the square root function valid. This gives us the domain for \(g\) as \(x \leq 2\). And as \(f(x)\) is a quadratic function with no restrictions, its domain is all real numbers. Hence the domain of the composite function \(f \circ g\) is \(x \leq 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function
The domain of a function refers to all the possible input values (commonly represented as \(x\)) that make the function work without any errors. Think of it like a list of permissible ingredients you can use in a recipe.

Here鈥檚 how to determine the domain:
  • Identify any mathematical operations that might restrict values, such as division by zero or taking the square root of a negative number.
  • Consider all real numbers, unless restrictions arise from these operations.
For example, with the function \(g(x) = \sqrt{2-x}\), the expression under the square root (\(2-x\)) must be zero or positive.
This calculation gives us \(2-x \geq 0\), leading to the domain \(x \leq 2\). This means \(x\) can be any number less than or equal to 2.
Understanding the domain is crucial because only the values within this set are valid inputs for the function.
Composite Functions
Composite functions, like the name suggests, are a combination of two or more functions. They allow us to apply one function to the results of another.
In mathematics, this is written as \(f(g(x))\) or \((f \circ g)(x)\). Here, you first calculate \(g(x)\) and then use this result as the input for function \(f(x)\).
Consider the given exercise where \(f(x) = x^2 + 1\) and \(g(x) = \sqrt{2-x}\):
  • First, solve \(g(x)\) which is \(\sqrt{2-x}\).
  • Take this result and substitute into \(f(x)\), resulting in \(f(g(x)) = (\sqrt{2-x})^2 + 1\).
  • Simplify the expression: \((\sqrt{2-x})^2 = 2-x\), so \(f(g(x)) = 2-x+1 = 3-x\).
With composite functions, it's important to carefully substitute and simplify to avoid mistakes. Remember, the domain of the composite function must satisfy both the inner function \(g(x)\) and the outer function \(f(x)\).
Square Root Function
The square root function is an interesting type due to its behavior with input values. It is represented as \(\sqrt{x}\), and it only accepts non-negative numbers because the square root of a negative number is not defined in the real number system.
While working with it, remember these key points:
  • The expression inside the square root sign must be zero or positive (\(x \geq 0\)).
  • The output of a square root function is always non-negative.
Take the function \(g(x) = \sqrt{2-x}\), for example. To ensure that \(g(x)\) is valid for real numbers, \(2-x\) must be greater than or equal to zero.
This restriction defines the domain of \(g(x)\) as \(x \leq 2\).
By understanding these nuances, you can effectively handle square root functions and their domains, ensuring mathematical accuracy in your solutions.

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Most popular questions from this chapter

Find all values of x satisfying the given conditions. $$f(x)=1-2 x, g(x)=3 x^{2}+x-1, \text { and }(f \circ g)(x)=-5$$

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation鈥檚 domain and range. $$x^{2}+(y-1)^{2}=1$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.

A company that sells radios has yearly fixed costs of \(\$ 600,000 .\) It costs the company \(\$ 45\) to produce each radio. Each radio will sell for \(\$ 65 .\) The company's costs and revenue are modeled by the following functions, where \(x\) represents the number of radios produced and sold: \(C(x)=600,000+45 x\) This function models the company's costs. \(R(x)=65 x\) This function models the company's revenue. Find and interpret \((R-C)(20,000),(R-C)(30,000),\) and \((R-C)(40,000)\)

The toll to a bridge costs \(\$ 6.00 .\) Commuters who frequently use the bridge have the option of purchasing a monthly discount pass for \(\$ 30.00 .\) With the discount pass, the toll is reduced to \(\$ 4.00 .\) For how many bridge crossings per month will the cost without the discount pass be the same as the cost with the discount pass? What will be the monthly cost for each option? (Section \(1.3,\) Example 3 )
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