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Chapter 1: Problem 32

Graph the function without using a graphing utility, and determine the domain and range. Write your answer in interval notation. $$g(t)=\sqrt{t-3}$$

Short Answer

Expert verified
The domain of the function \( g(t) = \sqrt{t-3} \) in interval notation is [3, +鈭�) and the range is [0, +鈭�)

Step by step solution

01

Determine the domain of the function

The square root function \( g(t) = \sqrt{t-3} \) is defined for all \( t \) that makes the quantity \( t-3 \) either positive or zero. Solving the inequality \( t - 3 \geq 0 \), there is \( t \geq 3 \). So the domain of the function in interval notation is [3, +鈭�)
02

Determine the range of the function

Since the function is a square root function and the square root of a number is always non-negative, the range of the function is all non-negative real numbers. It means the least value the function takes is 0 and can go up to infinity. So in interval notation, the range of the function is [0, +鈭�)
03

Graph the function

To graph the function, pick a number in the domain, say 4 and compute the value of the function. \( g(4) = \sqrt{4-3}= \sqrt{1}= 1 \). Thus, the point (4,1) is on the graph. The function starts from (3,0) and gently increases as \( t \) increases. This leads to a graph which starts from the point (3, 0) and goes up to infinity.

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Key Concepts

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

Square Root Function
The square root function is a type of function where the output is the square root of the input expression. Mathematically, this is expressed as \( f(x) = \sqrt{x} \). In the context of the given exercise, we are dealing with the function \( g(t) = \sqrt{t-3} \). This function only produces values that are the square roots of non-negative numbers, since the square root of a negative number is not defined in the set of real numbers.
This means that we only consider positive values or zero when computing the square roots. As the input \( t \) increases, the output of the function also increases, showing that the function has a gentle upward trajectory after it starts.
Function Domain
The domain of a function refers to all the possible input values that can be substituted into the function without resulting in an undefined or meaningless expression. For the square root function \( g(t) = \sqrt{t-3} \), the expression under the square root, \( t-3 \), must be greater than or equal to zero.
This is because the square root of negative numbers does not exist in the realm of real numbers. Solving the inequality \( t-3 \geq 0 \), we find that \( t \geq 3 \). Therefore, the domain of the function, in interval notation, is [3, +鈭�). This means any real number 3 or greater can be used as an input for this function.
Function Range
The range of a function is the set of possible output values, which are produced by the function from the domain values. For the square root function \( g(t) = \sqrt{t-3} \), the output of the function is always zero or positive, since the square root of a real number cannot be negative.
Thus, the smallest value that \( g(t) \) can take is 0, and as \( t \) increases, \( g(t) \) also increases without bound. Consequently, the range of the function in interval notation is [0, +鈭�), signifying that \( g(t) \) outputs every non-negative real number.

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Most popular questions from this chapter

In this set of exercises, you will use absolute value to study real-world problems. A ruler measures an object with an uncertainty of \(\frac{1}{16}\) inch. If a pencil is measured to be 8 inches, use absolute value notation to write an inequality for the range of possible lengths of the pencil.

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} 3, & -2 \leq x \leq 1 \\ -x^{2}, & 12 \end{array}\right.$$

This set of exercises will draw on the ideas presented in this section and your general math background. Show that \(|x-k|=|k-x|\), where \(k\) is any real number.

Solve the inequality. Express your answer in interval notation. $$-2 \leq 2 x+1 \leq 3$$

\(f(x)=5[[x]]-2\) (Hint: The greatest integer function is found under the option INT in a graphing calculator.)
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