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Chapter 1: Problem 63

Graph, using your grapher, and estimate the domain of each function. Confirm algebraically. $$ f(x)=\frac{\sqrt{x}}{x-5} $$

Short Answer

Expert verified
Domain: \([0, 5) \cup (5, \infty)\).

Step by step solution

01

Identify the Function

The function given is \( f(x) = \frac{\sqrt{x}}{x-5} \). We need to determine where this function is defined by analyzing the numerator and the denominator.
02

Determine Domain from Square Root

For the square root \( \sqrt{x} \) to be defined, \( x \) must be greater than or equal to 0. Therefore, \( x \geq 0 \).
03

Determine Domain from Denominator

For the function to be defined, the denominator cannot be zero. Thus, \( x - 5 eq 0 \), which implies \( x eq 5 \).
04

Combine Restrictions

Combining the conditions from Steps 2 and 3, the domain of the function is all \( x \) values such that \( x \geq 0 \) and \( x eq 5 \). In interval notation, this is \([0, 5) \cup (5, \infty)\).
05

Confirm with Graphing Tool

Graph \( f(x) = \frac{\sqrt{x}}{x-5} \) using a graphing calculator or software. Observe the graph to confirm that it is undefined at \( x = 5 \) and starts from \( x = 0 \) onwards, except at \( x = 5 \).
06

Review and Conclude

Review the algebraic restrictions and graph observations to confirm consistency. The domain \([0, 5) \cup (5, \infty)\) is correct.

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Key Concepts

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

Interval Notation
Interval notation is a convenient way to express the domain and range of functions. It uses intervals to describe which parts of the number line satisfy the conditions of a function.
For example, consider the function from the exercise:
  • The function is defined for all real numbers except certain restricted values.
  • It must satisfy both the conditions from the numerator (square root function) and the denominator.
An interval notation such as \([0, 5) \cup (5, \infty)\) captures these restrictions efficiently.
Here, \([0, 5)\) means from 0 to just below 5, including 0 but not 5. The union symbol \(\cup\) combines this interval with \((5, \infty)\), which includes all values greater than 5 up to infinity.
  • Use square brackets \([ ]\) to denote inclusion of an endpoint, meaning the value is within the interval.
  • Parentheses \(( )\) denote exclusion, meaning the value is not part of the interval.
Therefore, interval notation provides a precise and efficient way to communicate the valid input values for a function.
Square Root Function
The square root function is fundamental in mathematics and influences the domain of functions involving square roots.
Its notation is \( \sqrt{x} \) and it is real only when \( x \geq 0 \).
This means for any number to have a real square root, it cannot be negative.
  • This restriction directly affects the domain of the given function \( f(x) = \frac{\sqrt{x}}{x-5} \).
  • In the context of the exercise, since the square root has to be real, \(x\) must be zero or more, i.e., \(x \geq 0\).
Square roots are often graphically represented as a curve that begins at the origin (0,0) and rises to the right.
As one of the basic algebraic functions, understanding its properties allows you to solve more complex algebra equations and expressions effectively. It's crucial in solving problems like the exercise, where identifying when the square root function is defined helps in pinpointing the function's domain.
Graphing Calculator
Graphing calculators are powerful tools for visualizing mathematical concepts. They allow a more intuitive understanding of algebraic functions by providing a visual representation of an equation.
When dealing with complex functions like \( f(x) = \frac{\sqrt{x}}{x-5} \), graphing helps confirm algebraic findings about the domain.
Here's how to use it effectively:
  • Input the function precisely using your graphing calculator to generate a graph.
  • Observe where the graph starts, which represents the smallest value of \(x\) included in the domain, in this case, \(x = 0\).
  • Identify any breaks or gaps in the graph. Here, the graph has a discontinuity at \(x = 5\), confirming algebraic analysis.
Graphing calculators aid in verifying restrictions and providing insights into the behavior of functions across different regions.
This visual confirmation complements algebraic work and fosters a deeper understanding of solving mathematical tasks.

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Most popular questions from this chapter

Cost A manufacturing firm has a daily cost function of \(C(x)=3 x+10,\) where \(x\) is the number of thousands of an item produced and \(C\) is in thousands of dollars. Suppose the number of items that can be manufactured is given by \(x=n(t)=3 t,\) where \(t\) is measured in hours. Find \((C \circ n)(t),\) and state what this means.

Diamond and colleagues \(^{90}\) studied the growth habits of the Atlantic croaker, one of the most abundant fishes of the southeastern United States. The mathematical model that they created for the ocean larva stage was given by the equation $$L(t)=0.26 e^{2.876\left[1-e^{-0.0623 t}\right]}$$ where \(t\) is age in days and \(L\) is length in millimeters. Graph this equation. Find the expected age of a 3 -mm-long larva algebraically.

In 1997, Fuller and coworkers \(^{44}\) at Texas A \& M University estimated the operating costs of cotton gin plants of various sizes. For the next-to-largest plant, the total cost in thousands of dollars was given approximately by \(C(x)=\) \(0.059396 x^{2}+22.7491 x+224.664,\) where \(x\) is the annual quantity of bales in thousands produced. Plant capacity was 30,000 bales. Revenue was estimated at \(\$ 63.25\) per bale. Using this quadratic model, find the break- even quantity, and determine the production level that will maximize profit.

Let \(f(x)=x^{2}+1 .\) Find \(f[f(1)]\) and \(f[f(x)]\).

Using your grapher, graph on the same screen \(y_{1}=\) \(f(x)=|x|\) and \(y_{2}=g(x)=\sqrt{x^{2}} .\) How many graphs do you see? What does this say about the two functions?
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