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Chapter 1: Problem 53

Find the domain of the function. $$g(y)=\sqrt{y-10}$$

Short Answer

Expert verified
The domain of the function \(g(y) = \sqrt{y-10}\) is \(y \geq 10\). This means that the function is defined for all y greater than or equal to 10.

Step by step solution

01

Understanding Function Domains

In mathematics, the domain of a function is the complete set of possible values of the independent variable, in the function. In this function, g(y), y is the independent variable. Thus, finding the domain of this function means determining all possible values for y that will output a real number.
02

Determine the restrictions of the function

In the given function \(g(y) = \sqrt{y-10}\), there is a square root operation. The square root of a number is defined only for non-negative numbers. Therefore, the expression under the square root, \(y-10\), must be greater than or equal to 0, to avoid imaginary results. This forms the restriction for y in this function.
03

Solve for y

Setting the expression under the square root greater than or equal to 0, gives the inequality \(y-10 \geq 0\). Solving for y, by adding 10 to both sides, results in \(y \geq 10\).

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Most popular questions from this chapter

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x,\) and graphically determine the domain of the function.

Write a sentence using the variation terminology of this section to describe the formula. Area of a triangle: \(A=\frac{1}{2} b h\)

Sketch the graph of the function. $$g(x)=-[[x]]$$

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?
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