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Chapter 1: Problem 46

Let \(f(x)=|x|, g(x)=x^{2}-4\) \(F(x)=\sqrt{x},\) and \(G(x)=1 /(x-2) .\) Determine the following composite functions and give their domains. $$F \circ g \circ g$$

Short Answer

Expert verified
Answer: The composite function \(F \circ g \circ g\) is \(\sqrt{(x^2 - 4)^2 - 4}\), and its domain is the set of all real numbers, or in interval notation, \((-\infty, \infty)\).

Step by step solution

01

Find \(g \circ g\)

To find the composite function \(g \circ g\), we substitute \(g(x)\) into itself: \(g(g(x))\). Given that \(g(x) = x^2 - 4\), we have: $$g(g(x)) = g(x^2 - 4) = (x^2 - 4)^2 - 4.$$
02

Find \(F \circ g \circ g\)

Next, we need to find the composite function \(F \circ g \circ g\). To do this, we substitute the result from Step 1 into \(F(x) = \sqrt{x}\): $$F(g(g(x))) = F((x^2 - 4)^2 - 4) = \sqrt{(x^2 - 4)^2 - 4}.$$ So, \(F \circ g \circ g = \sqrt{(x^2 - 4)^2 - 4}\).
03

Determine the domain of \(F \circ g \circ g\)

The domain of a function is the set of all possible input values (in this case, \(x\)) for which the function is defined. Since we have a square root in our composite function, we need to ensure that the argument \((x^2 - 4)^2 - 4\) is greater than or equal to zero (as square root of a negative number is not defined in real numbers): $$(x^2 - 4)^2 - 4 \geq 0.$$ This inequality is always true because any squared term (in this case, \((x^2 - 4)^2\)) is always non-negative, and subtracting 4 from it does not make it less than 0. So, the domain of \(F \circ g \circ g\) is the set of all real numbers, or in interval notation, \((-\infty, \infty)\).

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Key Concepts

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

Function Composition
Function composition is the process of combining two or more functions to create a new function. In simpler terms, it's about feeding the output of one function into another. For example, if we have two functions, say \(f(x)\) and \(g(x)\), then the composition \(f \circ g\) means you apply \(g\) first and then \(f\) on the result of \(g\).

Here's a short breakdown of how we dealt with the original problem:
  • Start with the first function \(g(x) = x^2 - 4\).
  • Combine \(g\) with itself to find \(g \circ g\), which means substituting \(g(x)\) with \(g(g(x))\).
  • Then, apply \(F(x) = \sqrt{x}\) on the result of \(g \circ g\) to achieve the composite function \(F \circ g \circ g\).
This process highlights the importance of understanding each function and carefully following the order of operations when performing function composition.
Domain of a Function
The domain of a function is the set of possible input values, typically represented by \(x\), for which the function can operate without running into undefined operations. Understanding the domain helps in determining the range of values that are valid inputs, thereby avoiding issues like division by zero or taking square roots of negative numbers.

In the exercise, we determined the domain of the composite function \(F \circ g \circ g\) by analyzing its innermost structure:
  • Since \(F(x) = \sqrt{x}\), the inputs must be non-negative. Therefore, we ensure \((x^2 - 4)^2 - 4 \geq 0\).
  • Upon solving, it's apparent that \((x^2 - 4)^2\) is inherently non-negative, making \((x^2 - 4)^2 - 4\) always non-negative as well.
Hence, the domain of this composite is all real numbers, \(( -\infty, \infty )\). Understanding and applying these rules ensures functions are correctly defined over their respective domains.
Square Roots
Square roots are mathematical operations that find a number whose square is equal to the given number. Simply put, if \(b\) is a square root of \(a\), then \(b^2 = a\). Square roots are particularly significant when dealing with composite functions that include them since the expression under the square root must always be non-negative in real number calculations.

When we see square roots in functions, typically expressed as \(\sqrt{x}\), it's crucial to assess what values of \(x\) make the expression under the radical non-negative:
  • If \(x\) is negative, \(\sqrt{x}\) is not viable in the set of real numbers.
  • In the composite \(F(g(g(x))) = \sqrt{(x^2 - 4)^2 - 4}\), we had to ensure the term inside the square root was non-negative.
By understanding this, you safeguard functions from undefined behavior, thereby ensuring function integrity and correctness.

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Most popular questions from this chapter

Population model A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function \(p(t)=150 \times 2^{t / 12},\) where \(t\) is the number of hours after the first observation. a. Verify that \(p(0)=150,\) as claimed. b. Show that the population doubles every \(12 \mathrm{hr}\), as claimed. c. What is the population 4 days after the first observation? d. How long does it take the population to triple in size? e. How long does it take the population to reach \(10,000 ?\)

Find all the inverses associated with the following functions and state their domains. $$f(x)=2 x /(x+2)$$

Without using a calculator, evaluate or simplify the following expressions. $$\csc ^{-1}(\sec 2)$$

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A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function \(Q(t)=a\left(1-e^{-t / c}\right),\) where \(t\) is measured in seconds, and \(a\) and \(c > 0\) are physical constants. The steady-state charge is the value that \(Q(t)\) approaches as \(t\) becomes large. a. Graph the charge function for \(t \geq 0\) using \(a=1\) and \(c=10\) Find a graphing window that shows the full range of the function. b. Vary the value of \(a\) holding \(c\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(a ?\) c. Vary the value of \(c\) holding \(a\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(c ?\) d. Find a formula that gives the steady-state charge in terms of \(a\) and \(c\).
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