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Chapter 4: Problem 1

Why is it important to determine the domain of \(f\) before graphing \(f ?\)

Short Answer

Expert verified
Answer: Determining the domain of a function before graphing it is important because it helps to identify any undefined or discontinuous points in the graph, which could lead to potential misunderstandings or incorrect conclusions about the function's behavior. The domain represents the set of input values for which the function is defined, ensuring that the graph is accurately interpreted.

Step by step solution

01

Understand the function and its domain

The domain of a function \(f\) is the set of input values (usually represented as \(x\)) for which the function is defined. It is important to determine the domain before graphing the function to ensure you correctly interpret the graph.
02

Identify possible restrictions on the domain

Some common domain restrictions are caused by denominators of a fraction being equal to zero, square roots of negative numbers, and logarithms of non-positive numbers. Identify any such restrictions in your function, and note them, as they'll help in determining the domain.
03

Determine the domain

Using the identified restrictions, determine the domain of the function \(f\). This could be expressed in interval notation or as a set of allowed values for the input variable \(x\).
04

Graph the function

With the domain established, you can now proceed to graph the function. While graphing, take note of any undefined or discontinuous points and make sure they correspond to the domain restrictions you've identified earlier.
05

Interpret the graph with respect to the domain

Once the function is graphed, analyze the behavior, and interpret the information, keeping the domain of the function in mind. Understanding the domain will help you to make accurate conclusions about the function's behavior and properties.

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Most popular questions from this chapter

Suppose \(f(x)=1 /(1+x)\) is to be approximated near \(x=0\). Find the linear approximation to \(f\) at 0 . Then complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 \cdot |\) approximation \(-\) exact \(|/|\) exact \(| .\) Comment on the behavior of the errors as \(x\) approaches 0 .

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{aligned}&f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(-2)=0 ; f^{\prime}(-1)=f^{\prime}(1)=0\\\&f^{\prime \prime}(2)=0 ; f^{\prime}(3)=0 ; f^{\prime \prime}(x)>0 \text { on }(4, \infty)\end{aligned}$$

Determine whether the following statements are true and give an explanation or counterexample. a. If \(f^{\prime}(x)>0\) and \(f^{\prime \prime}(x)<0\) on an interval, then \(f\) is increasing at a decreasing rate on the interval. b. If \(f^{\prime}(c)>0\) and \(f^{\prime \prime}(c)=0,\) then \(f\) has a local maximum at \(c\) c. Two functions that differ by an additive constant both increase and decrease on the same intervals. d. If \(f\) and \(g\) increase on an interval, then the product \(f g\) also increases on that interval. e. There exists a function \(f\) that is continuous on \((-\infty, \infty)\) with exactly three critical points, all of which correspond to local maxima.

Determine whether the following statements are true and give an explanation or counterexample. a. \(F(x)=x^{3}-4 x+100\) and \(G(x)=x^{3}-4 x-100\) are antiderivatives of the same function. b. If \(F^{\prime}(x)=f(x),\) then \(f\) is an antiderivative of \(F\) c. If \(F^{\prime}(x)=f(x),\) then \(\int f(x) d x=F(x)+C\) d. \(f(x)=x^{3}+3\) and \(g(x)=x^{3}-4\) are derivatives of the same function. e. If \(F^{\prime}(x)=G^{\prime}(x),\) then \(F(x)=G(x)\)

a. For what values of \(b>0\) does \(b^{x}\) grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) b. Compare the growth rates of \(e^{x}\) and \(e^{a x}\) as \(x \rightarrow \infty,\) for \(a>0\).
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