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Chapter 1: Problem 18

Find the domain, range, and all zeros/intercepts, if any, of the functions. \(f(x)=\frac{1}{\sqrt{x-9}}\)

Short Answer

Expert verified
Domain is \((9, \infty)\), range is \((0, \infty)\). No zeros or intercepts.

Step by step solution

01

Identify Domain Restrictions

The function involves a square root and a division. The expression under the square root, \( x - 9 \), must be positive to avoid square root of a negative number. This means \( x - 9 > 0 \), so \( x > 9 \). Additionally, the square root must remain non-zero in the denominator to avoid division by zero. Thus, the domain of \( f \) is \( x > 9 \).
02

Determine the Domain

Based on the restrictions, the domain of the function \( f(x) = \frac{1}{\sqrt{x-9}} \) is \( (9, \infty) \), meaning that \( x \) can take any value greater than 9.
03

Find the Range

The function \( f(x) = \frac{1}{\sqrt{x-9}} \) outputs positive values only because both the numerator (1) and the square root (positive for \( x > 9 \)) in the denominator are positive. As \( x \to 9^+ \), \( f(x) \to \infty \); as \( x \to \infty \), \( f(x) \to 0^+ \). Thus, the range is \( (0, \infty) \).
04

Determine Zeros and Intercepts

A function has a zero where \( f(x) = 0 \). For \( f(x) = \frac{1}{\sqrt{x-9}} \), there is no \( x \) that makes \( f(x) = 0 \), because \( \frac{1}{\text{anything}} eq 0 \). The function also has no x-intercepts since it never touches the x-axis. The function also does not have a y-intercept, as there is no valid substitution for \( x = 0 \) within the domain.

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Key Concepts

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

Domain
Understanding the domain of a function is crucial because it tells us all the possible input values that the function can accept. For the function \( f(x) = \frac{1}{\sqrt{x-9}} \), we have to consider two things:
  • The expression inside the square root, \( x-9 \), must be positive since the square root of a negative number is undefined in the set of real numbers.
  • Additionally, because this expression is in the denominator, it cannot be zero, to avoid division by zero.
By solving \( x-9 > 0 \), we find that \( x > 9 \). Thus, the domain of the function is all real numbers greater than 9, which is mathematically represented as \( (9, \infty) \). This means you can only input numbers larger than 9 into this function.
Range
The range of a function is the set of possible output values it can produce. Exploring the function \( f(x) = \frac{1}{\sqrt{x-9}} \), we notice a few characteristics:
  • The output value will always be positive because the numerator is a constant positive number (1), and the square root in the denominator is always positive for \( x > 9 \).
  • As \( x \) approaches 9 from the right (getting close to 9 but never equaling it), the denominator gets very small, causing \( f(x) \) to increase towards infinity.
  • Conversely, as \( x \) approaches infinity, the function value \( f(x) \) approaches zero, but never actually reaches it.
These insights lead us to conclude that the range of the function is all positive numbers, denoted as \( (0, \infty) \). This tells us that no matter what input you choose from the domain, the function will return a positive output.
Intercepts
Intercepts are the points where a function crosses the x-axis or y-axis. For the function \( f(x) = \frac{1}{\sqrt{x-9}} \), it's essential to analyze the intercepts:
  • **X-intercepts:** These occur when the function output is zero. For \( f(x) \) to be zero, it would imply \( \frac{1}{something} = 0 \), which is impossible. Thus, there are no x-intercepts for this function.
  • **Y-intercepts:** These happen when the function鈥檚 input \( x \) is zero. Plugging in \( x = 0 \) into \( f(x) \) results in a negative under the square root (since \( x - 9 = -9 \)), which is not allowed. Therefore, there are no y-intercepts, as x cannot be zero or any number less than 9.
Understanding intercepts helps us know the points where the function changes sign or crosses the axes, and for \( f(x) = \frac{1}{\sqrt{x-9}} \), the lack of intercepts aligns with its domain and range.

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