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Chapter 0: Problem 24

Find the domain of the function. $$ f(x)=\frac{\sqrt[3]{x^{2}-x+1}}{x^{2}+1} $$

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{\sqrt[3]{x^{2}-x+1}}{x^{2}+1}\) is all real numbers, denoted as \((-鈭�, 鈭�)\).

Step by step solution

01

Examine the numerator

The numerator consists of a cubic root function, which is defined for all real numbers. Therefore, there are no restrictions on x-values from the numerator. Step 2: Examine the denominator
02

Examine the denominator

We need to check if there are any x-values that make the denominator equal to zero. Given the denominator is \(x^2 + 1\), let's find the values of x for which this expression is equal to zero: $$x^2 + 1 = 0$$ $$x^2 = -1$$ We can see that there is no real value of x for which \(x^2 = -1\). Therefore, the denominator is never zero for any real value of x. Step 3: Determine the domain
03

Determine the domain

Since there are no restrictions on x-values from the numerator and the denominator is never zero for any real value of x, the domain of the function is all real numbers. In interval notation, the domain of the function \(f(x)=\frac{\sqrt[3]{x^{2}-x+1}}{x^{2}+1}\) is written as: $$(-\infty, \infty)$$

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Most popular questions from this chapter

Determine whether the function is one-to-one. $$ f(x)=-x^{4}+16 $$

Let \(f(x)=\left(1+\frac{1}{x}\right)^{x}\), where \(x>0\). a. Plot the graph of \(f\) using the window \([0,10] \times[0,3]\), and then using the window \([0,100] \times[0,3] .\) Does \(f(x)\) appear to approach a unique number as \(x\) gets larger and larger? b. Use the evaluation function of your graphing utility to fill in the accompanying table. Use the table of values to estimate, accurate to five decimal places, the number that \(f(x)\) seems to approach as \(x\) increases without bound. Note: We will see in Section \(2.8\) that this number, written \(e\), is given by \(2.71828 \ldots\)

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\sqrt{x}, \quad y=2 \sqrt{x-1}+1\)

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=x^{2}, \quad y=(x-2)^{2}\)

Find the inverse of \(f .\) Then sketch the graphs of \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{3}+1 $$
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