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Chapter 2: Problem 55

Find the domain of the function. $$f(x)=\frac{3}{\sqrt{x-4}}$$

Short Answer

Expert verified
The domain of the function is \((4, \infty)\).

Step by step solution

01

Identify the Denominator

The given function is \( f(x)= rac{3}{ ext{expression}} \). Identify the denominator. Here, it is \( \sqrt{x-4} \).
02

Determine Valid Conditions for Roots

For \( \sqrt{x-4} \) to be a real number, the expression inside the square root must be non-negative. Therefore, set up the inequality: \( x-4 \geq 0 \).
03

Solve the Inequality

Solve the inequality \( x-4 \geq 0 \). Adding 4 to both sides, we get \( x \geq 4 \). This means that "\(x\)" can take any value greater than or equal to 4.
04

Evaluate Fraction Division

In a fraction, the denominator cannot be zero. Thus, \( \sqrt{x-4} eq 0 \). But since \( \sqrt{x-4} eq 0 \) translates to \( x eq 4 \), we already cover this with \( x \geq 4 \).
05

Combine Results

Considering \( x \geq 4 \) from Step 3 and the fraction condition that eliminates only \( x = 4 \), the domain is \( x > 4 \).
06

Express the Domain

Convert the domain condition into interval notation. Since \( x \) must be greater than 4, the domain in interval notation is \((4, \infty)\).

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Key Concepts

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

Square Root Function
A square root function involves an expression that includes a square root. In our case, the function is \( f(x) = \frac{3}{\sqrt{x-4}} \). The presence of the square root \( \sqrt{x-4} \) indicates that we need to consider the domain of the function carefully. The expression inside the square root, \( x-4 \), must be non-negative because square roots of negative numbers are not real. Therefore, we set up the condition \( x-4 \geq 0 \). This inequality ensures the expression under the square root is at least zero, making it a real, non-negative number.
  • Square roots only accept non-negative inputs.
  • The expression inside the root dictates the constraints on \( x \).
Understanding the square root helps us determine the valid range of \( x \) for the function to be defined.
Inequalities
Inequalities are mathematical expressions used to define a range of values. In this context, they help us identify where the function is defined. To find the domain of the function with a square root, we derive an inequality from the expression \( \sqrt{x-4} \). Specifically, we identify \( x-4 \geq 0 \). Solving this inequality requires simple algebra. When we add 4 to both sides, we get \( x \geq 4 \).
  • Inequalities allow evaluation of conditions for mathematical expressions.
  • Algebraic manipulation helps solve these inequalities.
This process ensures the inside of the square root is non-negative, which is especially important in real numbers.
Interval Notation
Interval notation is a format used to precisely describe a set of numbers, often concerning the domain of functions. Once the valid \( x \)-values are identified from inequalities, interval notation beautifully summarizes the range. In this example, we translate \( x > 4 \) into interval notation, rendering it as \((4, \infty)\).The parentheses \((4, \infty)\) indicate that 4 is not included, but any number greater than 4 is part of the domain. The \( \infty \) symbol is always accompanied by a parenthesis because infinity is a concept, not a tangible number you can "reach."
  • Interval notation efficiently presents ranges.
  • Parentheses or brackets show inclusion/exclusion of end points.
This notation offers a concise way to express which \( x \)-values the function can accept.

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Most popular questions from this chapter

Area of a Box An open box with a square base is to have a volume of \(12 \mathrm{ft}^{3}\) (a) Find a function that models the surface area of the box. (b) Find the box dimensions that minimize the amount of material used.

Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-4,6]\) by \([-4,4]\) (a) \(y=x^{6}\) (b) \(y=\frac{1}{3} x^{6}\) (c) \(y=-\frac{1}{3} x^{6}\) (d) \(y=-\frac{1}{3}(x-4)^{6}\)

Express the function in the form \(f \circ g \circ h\) $$G(x)=\frac{2}{(3+\sqrt{x})^{2}}$$

Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$f(x)=x+\frac{1}{x}$$

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. \(f(x)=1 / x^{n}\) (a) \(n=1,3 ;[-3,3]\) by \([-3,3]\) (b) \(n=2,4 ;[-3,3]\) by \([-3,3]\) (c) How does the value of \(n\) affect the graph?
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