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Chapter 1: Problem 17

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator.] $$ f(x)=x^{2 / 3} ; \text { find } f(-8) $$

Short Answer

Expert verified
a. \(f(-8)=4\); b. Domain: \((-\infty, \infty)\); c. Range: \([0, \infty)\).

Step by step solution

01

Evaluate the expression

To evaluate the expression \( f(x) = x^{2/3} \) at \( x = -8 \), we compute \( f(-8) = (-8)^{2/3} \). This can be broken down as \( ((-8)^{1/3})^2 \). The cube root of \(-8\) is \(-2\) because \((-2)^3 = -8\). Therefore, \((-2)^2 = 4\). Thus, \( f(-8) = 4 \).
02

Find the domain of the function

The domain of \( f(x) = x^{2/3} \) is all real numbers. The reason is that both the cube root function and the squaring function are defined for all real numbers. Hence, the domain is \( (-\infty, \infty) \).
03

Find the range of the function

To find the range, note that for \( x^{2/3} \), the output is always non-negative because any real number raised to the power of \( 2/3 \) will be non-negative. Thus, the range is \([0, \infty)\).

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Key Concepts

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

Evaluating Functions
Evaluating a function means finding the output value when an input value is plugged into a function. It鈥檚 like putting an ingredient into a blender and seeing what comes out. To evaluate the function given in the exercise, which is \( f(x) = x^{2/3} \), we need to find \( f(-8) \).

Here鈥檚 how it鈥檚 done step by step:
  • First, replace \( x \) with \( -8 \) in the function: \( f(-8) = (-8)^{2/3} \).
  • Calculate the cube root of \(-8\). The cube root of \(-8\) is \(-2\), because when you multiply \(-2\) by itself three times, you get \(-8\): \((-2) \times (-2) \times (-2) = -8\).
  • Next, square \(-2\). This means: \((-2)^2 = 4\).
So, after evaluating the function at \( x = -8 \), we find that \( f(-8) = 4 \). It's important to break down the steps to make sure you are calculating the powers and roots in the right order.
Domain of a Function
The domain of a function is all the possible input values (usually \( x \)) that will give a valid output when substituted into the function. Think of it as the set of all possible ingredients you can put into the blender.

For the function \( f(x) = x^{2/3} \), we consider both the cube root and the square operation:
  • The cube root function \( x^{1/3} \) can accept any real number because you can take the cube root of both positive and negative numbers, as well as zero.
  • The square function \((...)^2\) also accepts any real number as input.
Since both operations accept all real numbers, the domain of \( f(x) = x^{2/3} \) is all real numbers. In interval notation, we write this as \((-\infty, \infty)\), meaning you can plug any real number into the function.
Range of a Function
The range of a function is all the possible output values (often \( y \)) the function can produce. It's like all the different smoothies you can make with the blender, depending on what you put in it.

For the function \( f(x) = x^{2/3} \), the range is determined by examining the possible outputs:
  • Note that when you take the cube root of any real number, the result can be negative, zero, or positive.
  • However, when you then square that result, the squaring operation changes any negative outcomes to positive. For example, \(-2^2 = 4\) and \(0^2 = 0\).
As a result, regardless of the initial sign of \( x \), the outcome of \((x^{1/3})^2 \) will always be non-negative. Therefore, the range of \( f(x) = x^{2/3} \) is \([0, \infty)\), including zero and extending to positive infinity.

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Most popular questions from this chapter

How do the graphs of \(f(x)\) and \(f(x)+10\) differ?

The following function expresses an income tax that is \(10 \%\) for incomes below $$\$ 5000$$, and otherwise is $$\$ 500$$ plus \(30 \%\) of income in excess of $$\$ 5000$$. \(f(x)=\left\\{\begin{array}{ll}0.10 x & \text { if } 0 \leq x<5000 \\ 500+0.30(x-5000) & \text { if } x \geq 5000\end{array}\right.\) a. Calculate the tax on an income of $$\$ 3000$$. b. Calculate the tax on an income of $$\$ 5000$$. c. Calculate the tax on an income of $$\$ 10,000$$. d. Graph the function.

Graph each function. $$ f(x)=|x-3|-3 $$

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

Use a graphing calculator to graph each piecewise nonlinear function on the window \([-2,10]\) by \([-5,5]\). Where parts of the graph do not touch, state which point is included and which is excluded. $$ f(x)=\left\\{\begin{array}{ll} 4-x^{2} & \text { if } x<3 \\ 2 x-11 & \text { if } 3 \leq x<7 \\ 8-x & \text { if } x \geq 7 \end{array}\right. $$
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