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Chapter 2: Problem 67

Consider \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\). Why are the domains of \(f\) and \(g\) different?

Short Answer

Expert verified
The domains of the functions \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\) are different due to the mathematical restrictions of the respective root types. The domain of the square root function \(f(x)\) requires the argument to be greater than or equal to zero (hence \(x \geq 2\)), while the cube root function \(g(x)\) can accept all real numbers as input.

Step by step solution

01

Find the domain of the function \(f(x)=\sqrt{x-2}\)

The function \(f(x)=\sqrt{x-2}\) represents a square root function. It is important to remember that a square root function is only defined for values that are greater than or equal to zero, since attempting to take the square root of a negative number yields an imaginary result, which is outside of the set of real numbers. Consequently, to find the domain of this function, set the expression under the square root to be greater than or equal to zero, \(x-2 \geq 0\). Solving for \(x\), we find that \(x \geq 2\). Thus, the domain of \(f(x)\) is \(x \geq 2\).
02

Find the domain of the function \(g(x)=\sqrt[3]{x-2}\)

The function \(g(x)=\sqrt[3]{x-2}\) represents a cube root function. Differently from square root functions, cube root functions can take negative values and still yield real number solutions. Hence, there are no restrictions on the domain and it encompasses all real numbers.
03

Compare the domains of \(f(x)\) and \(g(x)\)

The reason for the difference in the domains between \(f(x)\) and \(g(x)\) is due to the mathematical restrictions associated with their respective root types. A square root function requires its argument to be non-negative (greater than or equal to 0), while a cube root function can accept all real numbers (including negative numbers) as input. Hence, the domain of \(f(x)\) is \(x \geq 2\) and the domain of \(g(x)\) is all real numbers.

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Most popular questions from this chapter

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is reflected in the \(x\) -axis, shifted two units to the left, and shifted one unit upward.

The number of horsepower \(H\) required to overcome wind drag on an automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Rewrite the horsepower function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\) Identify the type of transformation applied to the graph of the horsepower function.

Find (a) \(f \circ g\) and (b) \(g \circ f\). . \(f(x)=\sqrt{x}, \quad g(x)=\sqrt{x}\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is vertically stretched by a factor of 4 .

Use a graphing utility to graph \(f\) for \(c=-2,0\), and 2 in the same viewing window. (a) \(f(x)=\frac{1}{2} x+c\) (b) \(f(x)=\frac{1}{2}(x-c)\) (c) \(f(x)=\frac{1}{2}(c x)\) In each case, compare the graph with the graph of \(y=\frac{1}{2} x\).
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