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Chapter 1: Problem 84

CAPSTONE Consider the functions \(f(x) = x^2\) and \(g(x) = \sqrt{x}\). (a) Find \((f/g)\) and its domain. (b) Find \((f \circ g)\) and \((g \circ f)\). Find the domain of each composite function. Are they the same? Explain.

Short Answer

Expert verified
The quotient function is \(x \sqrt{x}\) with a domain of \([0, +\infty)\). The composite function \(f \circ g\) is \(x\) with a domain of \([0, +\infty)\), while the composite function \(g \circ f\) is \(|x|\) with a domain of \(-\infty, +\infty\). The domains of the two composite functions are not the same.

Step by step solution

01

find the quotient function

To find the quotient of the two functions \(f/g\), simply divide \(f(x)\) by \(g(x)\). \[ (f/g)(x) = \frac{x^2}{\sqrt{x}} \] After simplifying this expression, you'll get \( (f/g)(x) = x \sqrt{x}\).
02

find the domain of the quotient function

The domain of \( (f/g)(x) \) is given by all the x-values where the function is defined. Since we cannot divide by zero or take the square root of a negative number, the domain of \( (f/g)(x) \) is \([0, +\infty)\).
03

find the composite functions

To find the composite functions, replace each function within the other. This provides \( (f \circ g)(x) = f(g(x)) = (\sqrt{x})^2 = x \) and \( (g \circ f)(x) = g(f(x)) = \sqrt{x^2} = |x|\).
04

find the domains of the composite functions

For \(f \circ g\), we have \(x \geq 0\) because we can't take the square root of a negative number. Hence domain of \(f \circ g\) is \([0, +\infty)\). For \(g \circ f\), since \(f(x) = x^2\) is always positive or zero, \(g(f(x)) = \sqrt{x^2} = |x|\) is defined for all x, so its domain is \(-\infty, +\infty\).
05

Compare the domains of the composite functions

We found that the domain of \(f \circ g\) is \([0, +\infty)\), while the domain of \(g \circ f\) is \(-\infty, +\infty\). So the domains of the two composite functions are not the same.

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Key Concepts

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

Domain of a Function
In mathematics, the domain of a function is the set of all possible input values (usually denoted as 'x') that the function can accept without any issues. Determining the domain involves identifying restrictions that would cause the function to break.

For our example functions, \(f(x) = x^2\) and \(g(x) = \sqrt{x}\), let's focus on the quotient \((f/g)(x)\). To find its domain, consider two rules:
  • You cannot divide by zero.
  • You cannot take the square root of a negative number.
Thus, the domain for \((f/g)(x) = x\sqrt{x}\) is \([0, +\infty)\). This means all non-negative numbers can be inputs, as any negative inputs would make \(g(x)\) undefined.

This concept of domain ensures your calculations stay accurate and functions return real, valid results.
Composite Functions
Composite functions are formed when one function is applied to the result of another function. This is often noted as \((f \circ g)(x)\) or \((g \circ f)(x)\).

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). This yields \((f \circ g)(x) = f(g(x)) = (\sqrt{x})^2 = x\).

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\), giving us \(g(f(x)) = \sqrt{x^2} = |x|\).

Each composite function may have a different domain:
  • \((f \circ g)(x)\) only works where \(x\) is non-negative, so its domain is \([0, +\infty)\).
  • \((g \circ f)(x)\) is defined for all real numbers since \(|x|\) covers both positive and negative values. Therefore, its domain is \((-\infty, +\infty)\).
Understanding how each function affects the other helps identify these domains.
Quotient of Functions
Dividing one function by another results in a quotient function, often represented as \(\frac{f(x)}{g(x)}\).

In the example given, forming the quotient \((f/g)(x)\) involves dividing \(f(x)\) by \(g(x)\), yielding:\[\frac{x^2}{\sqrt{x}} = x\sqrt{x}\]

The rules remain the same: avoid division by zero and invalid operations such as taking the square root of negative numbers. That鈥檚 why the domain for \((f/g)(x) = x\sqrt{x}\) is \([0, +\infty)\).

Quotient functions can be tricky due to these limitations, but careful attention to rules and simplifications helps navigate these potential pitfalls effectively.

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Most popular questions from this chapter

THINK ABOUT IT In Exercises 77-86, restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1}\). State the domains and ranges of \(f\) and \(f^{-1}\). Explain your results. (There are many correct answers.) \(f(x) = \frac{1}{2}x^2 - 1\)

RESISTANCE In Exercises 77 and 78, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms?

In Exercises 63-76, determine whether the function has an inverse function. If it does, find the inverse function. \(p(x) = -4\)

ECOLOGY In Exercises 75 and 76, use the fact that the diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. A stream of velocity \(v\) can move particles of diameter \(d\) or less. By what factor does \(d\) increase when the velocity is doubled?

DATA ANALYSIS: LIGHT INTENSITY A light probe is located \(x\) centimeters from a light source, and the intensity \(y\) (in microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs \((x, y)\). \((30, 0.1881)\) \((34, 0.1543)\) \((38, 0.1172)\) \((42, 0.0998)\) \((48, 0.0775)\) \((50, 0.0645)\) A model for the data is \(y = 262.76/x^{2.12}\) (a) Use a graphing utility to plot the data points and the model in the same viewing window. (b) Use the model to approximate the light intensity 25 centimeters from the light source.
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