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Chapter 1: Problem 121

Given \(f(x)=x^{2},\) is \(f\) the independent variable? Why or why not?

Short Answer

Expert verified
No, \(f\) is not the independent variable. In the function \(f(x)\), the independent variable is \(x\) because its value can change freely. The result of \(f(x) = x^{2}\) changes based on the value of \(x\), making \(f(x)\) the dependent variable.

Step by step solution

01

Understanding the notation

In the function \(f(x) = x^{2}\), \(x\) and \(f(x)\) are two different things. Here, \(x\) is the variable, and \(f(x)\) is the function which is equal to the square of \(x\).
02

Identifying the independent variable

In this context, \(x\) is the independent variable because it can take any value freely, and its value is not dependent on any other variable.
03

Identifying the dependent variable

The quantity \(f(x)\), which is equal to \(x^{2}\), is the dependent variable here because its value depends on the chosen value of \(x\).

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Most popular questions from this chapter

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{1}{x^{2}} $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x-3}{x+2} $$

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=\frac{1}{4} $$

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$
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