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Chapter 1: Problem 14

Find the domain for each of the following functions: a. \(f(x)=300.4+3.2 x\) b. \(g(x)=\frac{5-2 x}{2}\) c. \(j(x)=\frac{1}{x+1}\) d. \(k(x)=3\) e. \(f(x)=x^{2}+3\)

Short Answer

Expert verified
a. \( (-\infty, \infty) \) b. \( (-\infty, \infty) \) c. \( (-\infty, -1) \cup (-1, \infty) \) d. \( (-\infty, \infty) \) e. \( (-\infty, \infty) \)

Step by step solution

01

- Finding the Domain of Function (a)

The function given is a linear function: \( f(x)=300.4+3.2 x \) A linear function is defined for all real numbers. Hence, the domain is \( (-\infty, \infty) \).
02

- Finding the Domain of Function (b)

The function given is a rational function: \( g(x)=\frac{5-2 x}{2} \) A rational function is defined except where the denominator is zero. The denominator here is 2, which is never zero. Therefore, the domain is \( (-\infty, \infty) \).
03

- Finding the Domain of Function (c)

The function given is a rational function: \( j(x)=\frac{1}{x+1} \) A rational function is defined except where the denominator is zero. Set the denominator equal to zero and solve: \( x+1=0 \) Therefore, \( x=-1 \) is excluded from the domain. The domain is \( (-\infty, -1) \cup (-1, \infty) \).
04

- Finding the Domain of Function (d)

The function given is a constant function: \( k(x)=3 \) A constant function is defined for all real numbers. Hence, the domain is \( (-\infty, \infty) \).
05

- Finding the Domain of Function (e)

The function given is a quadratic function: \( f(x)=x^{2}+3 \) A quadratic function is defined for all real numbers. Hence, the domain is \( (-\infty, \infty) \).

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Key Concepts

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

linear function
A linear function is one of the simplest types of functions. It takes the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants, and \( x \) represents the variable.
Here, \( m \) is the slope of the line and \( b \) is the y-intercept, or where the line crosses the y-axis.
The domain of a linear function is all real numbers, denoted by \( (-\infty, \infty) \). This means no matter what real number you plug in for \( x \), you will always get a corresponding y-value.
rational function
A rational function is formed by the ratio of two polynomials. It takes the form \( g(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).
The domain of a rational function includes all real numbers except where the denominator equals zero, since division by zero is undefined. To find the domain, set the denominator equal to zero and solve for \( x \).
For example, in \( j(x) = \frac{1}{x+1} \), setting \( x+1 = 0 \) gives \( x = -1 \). So, the domain is all real numbers except \( x = -1 \). This means \( (-\infty, -1) \cup (-1, \infty) \).
constant function
A constant function has the form \( k(x) = c \), where \( c \) is a constant.
No matter what value of \( x \) you input, the output will always be \( c \). This type of function is represented by a horizontal line on the graph.
The domain of a constant function is all real numbers, \( (-\infty, \infty) \), because the value of \( x \) can be anything without affecting the output.
quadratic function
A quadratic function has the form \( f(x) = ax^{2} + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
This type of function creates a parabola when graphed.
The domain of a quadratic function is all real numbers denoted by \( (-\infty, \infty) \). This is because you can plug any real number into the quadratic equation and get a corresponding output.
domain of a function
The domain of a function refers to all the possible input values (\( x \) values) that the function can accept.
To find the domain:
  • For linear, constant, and quadratic functions, the domain is all real numbers (\( (-\infty, \infty) \)).
  • For rational functions, exclude values that make the denominator zero.
Identifying the domain of a function is important because it tells us the set of all possible inputs we can use for the function to produce real and valid outputs.

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Most popular questions from this chapter

The accompanying table gives the ages of students in a mathematics class. Ages of Students \begin{tabular}{cc} \hline Age Interval & Frequency Count \\ \hline \(15-19\) & 2 \\ \(20-24\) & 8 \\ \(25-29\) & 4 \\ \(30-34\) & 3 \\ \(35-39\) & 2 \\ \(40-44\) & 1 \\ \(45-49\) & 1 \\ \hline Total & 21 \\ \hline \end{tabular} a. Use this information to estimate the mean age of the students in the class. Show your work. (Hint: Use the mean age of each interval.) b. What is the largest value the actual mean could have? The smallest? Why?

The Greek letter \(\Sigma\) (called sigma) is used to represent the sum of all of the terms of a certain group. Thus. \(a_{1}+a_{2}+a_{3}+\cdots+a_{n}\) can be written as $$ \sum_{i=1}^{n} a_{i} $$ which means to add together all of the values of \(a_{i}\) from \(a_{1}\) to \(a_{\mathrm{an}}\) a. Using \(\Sigma\) notation, write an algebraic expression for the mean of the five numbers \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) b. Using \(\Sigma\) notation, write an algebraic expression for the mean of the \(n\) numbers \(t_{1}, t_{2}, t_{3}, \ldots ., t_{n^{*}}\) c. Evaluate the following sum: $$ \sum_{k=1}^{5} 2 k $$

Determine the domain of each of the following functions. Explain your answers. \(f(x)=4 \quad g(x)=3 x+5 \quad h(x)=\frac{x-1}{x-2}\) \(F(x)=x^{2}-4 \quad G(x)=\frac{x-1}{x^{2}-4} \quad H(x)=\sqrt{x-2}\)

An article titled "Venerable Elders" (The Economist, July 24, 1999 ) reported that "both Democratic and Republican images are selective snapshots of a reality in which the median net worth of households headed by Americans aged 65 or over is around double the national average-but in which a tenth of such households are also living in poverty." What additional statistics would be useful in forming an opinion on whether elderly Americans are wealthy or poor compared with Americans as a whole?

Given here is a table of salaries taken from a survey of recent graduates (with bachelor degrees) from a well-known university in Pittsburgh. $$ \begin{array}{cc} \hline \begin{array}{c} \text { Salary } \\ \text { (in thousands) } \end{array} & \begin{array}{c} \text { Number of Graduates } \\ \text { Receiving Salary } \end{array} \\ \hline 21-25 & 2 \\ 26-30 & 3 \\ 31-35 & 10 \\ 36-40 & 20 \\ 41-45 & 9 \\ 46-50 & 1 \\ \hline \end{array} $$ a. How many graduates were surveyed? \(\mathbf{b}\). Is this quantitative or qualitative data? Explain. c. What is the relative frequency of people having a salary between \(\$ 26,000\) and \(\$ 30,000 ?\) d. Create a histogram of the data.
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