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Chapter 1: Problem 224

For the following exercises, use each set of functions to find \(f(g(h(x))) .\) Simplify your answers. Given functions \(q(x)=\frac{1}{\sqrt{x}}\) and \(h(x)=x^{2}-9,\) state the domain of each of the following functions using interval notation. a. \(\frac{q(x)}{h(x)}\) b. \(q(h(x))\) c. \(\quad h(q(x))\)

Short Answer

Expert verified
a. \((0, 3) \cup (3, \infty)\); b. \((-\infty, -3) \cup (3, \infty)\); c. \((0, \infty)\).

Step by step solution

01

Determine the Domain of q(x)

The function \(q(x) = \frac{1}{\sqrt{x}}\) is defined for all \(x > 0\) since the square root function requires the argument to be positive. Therefore, the domain of \(q(x)\) in interval notation is \((0, \infty)\).
02

Determine the Domain of h(x)

The function \(h(x) = x^2 - 9\) is a polynomial, which is defined for all real numbers. Thus, the domain of \(h(x)\) in interval notation is \((-\infty, \infty)\).
03

Find the Domain of \(\frac{q(x)}{h(x)}\)

The domain of \(\frac{q(x)}{h(x)}\) is determined by the intersection of the domains of \(q(x)\) and \(h(x)\), excluding points where \(h(x) = 0\). Since \(q(x)\) is defined for \(x > 0\) and \(h(x) = 0\) when \(x^2 - 9 = 0\) (i.e., \(x = \pm3\)), the domain is \((0, 3) \cup (3, \infty)\).
04

Find the Domain of \(q(h(x))\)

Since \(q(x) = \frac{1}{\sqrt{x}}\) requires that \(x > 0\), it implies \(h(x) > 0\). Thus, \(x^2 - 9 > 0\). This inequality holds for \(x > 3\) and \(x < -3\). Therefore, the domain is \((-\infty, -3) \cup (3, \infty)\).
05

Find the Domain of \(h(q(x))\)

The function \(h(q(x)) = \left(\frac{1}{\sqrt{x}}\right)^2 - 9\) simplifies to \(\frac{1}{x} - 9\). As \(q(x)\) is defined for \(x > 0\), the domain of \(h(q(x))\) remains the same as \((0, \infty)\).

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Key Concepts

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

Function Domain
The domain of a function is a set of all possible input values (often represented by variable \(x\)) for which the function is defined. It's like the boundaries within which a function "lives". For example, if a function involves a square root, like \(q(x) = \frac{1}{\sqrt{x}}\), you can only input positive numbers, since square roots of negative numbers aren't real. The same goes for division by zero, which must be avoided.Here's a simpler guide to find a function's domain:
  • If the function is a basic polynomial, like \(h(x) = x^2 - 9\), it usually covers all real numbers, \((-\infty, \infty)\).
  • If it's a fractional or square root function, check for values that make the function undefined, like \(x = 0\) in \(q(x)\), making its domain \((0, \infty)\).
  • Combine the rules if functions are composed, such as in \(f(g(h(x)))\).
Understanding the domain is crucial because it tells you where the function works properly without leading to mathematical absurdities.
Interval Notation
Interval notation is a neat way to describe the domain or range of a function using intervals. It specifies the set of numbers that are included (or excluded) from a function.In interval notation:
  • Parentheses, like \((a, b)\), indicate that the endpoints \(a\) and \(b\) are not included in the interval.
  • Brackets, like \([a, b]\), signal that the endpoints are included.
  • Infinity signs \((\infty, -\infty)\) are always paired with parentheses, since infinity isn鈥檛 an actual number we can reach.
For instance, in the function \(q(x) = \frac{1}{\sqrt{x}}\), its domain isn鈥檛 \([-3, 3]\) because \(x\) has to be greater than zero, hence \((0, \infty)\). Knowing interval notation is like knowing how to read a map or blueprint. It guides you to where the function is operational.
Polynomial Functions
Polynomial functions are expressions involving variables raised to whole number powers. They look like \(h(x) = x^2 - 9\), where the highest degree (in this case, \(x^2\)) dictates much of the function's behavior. Key characteristics of polynomials:
  • They are defined for all real numbers, so the domain is always \((-\infty, \infty)\).
  • The graph of a polynomial is a smooth, continuous curve with no breaks or holes.
  • They can have multiple roots or zeros, where the function鈥檚 value is zero (for \(x^2 - 9\), roots are at \(x=3\) and \(x=-3\)).
These properties make polynomials flexible and easy to work with, which is why they frequently appear in mathematical models and real-world applications.
Rational Functions
Rational functions are ratios of two polynomials, like the expression \(\frac{q(x)}{h(x)}\). These can be tricky as they introduce restrictions based on their denominators.Considerations with rational functions:
  • The function is undefined wherever the denominator is zero. You must exclude these values from the domain.
  • For example, \(\frac{q(x)}{h(x)} = \frac{\frac{1}{\sqrt{x}}}{x^2-9}\) excludes points that make \(h(x) = 0\), like \(x=\pm3\).
  • The domain is the intersection of possible values from the numerator and denominator. For \(\frac{q(x)}{h(x)}\), it results in \((0, 3) \cup (3, \infty)\).
Rational functions can model complex real-world scenarios but require careful handling to identify all restrictions and ensure meaningful results.

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Most popular questions from this chapter

Given \(f(x)=2 x^{2}+1\) and \(g(x)=3 x-5,\) find the following: \(\begin{array}{ll}{\text { a. }} & {f(g(2))} \\ {\text { b. }} & {f(g(x))} \\\ {\text { c. }} & {g(f(x))} \\ {\text { d. }} & {(g \circ g)(x)} \\\ {\text { e. }} & {(f \circ f)(-2)}\end{array}\)

Write a formula for the function obtained when the graph of \(f(x)=\frac{1}{x^{2}}\) is shifted up 2 units and to the left 4 units.

Write a formula for the function obtained when the graph of \(f(x)=\sqrt{x}\) is shiffed to the left 2 units.

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. $$ g(x)=5(x+3)^{2}-2 $$

For the following exercises, determine whether the function is odd, even, or neither. $$ h(x)=2 x-x^{3} $$
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