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Chapter 2: Problem 29

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$f(x)=\frac{4 x^{2}-1}{x^{2}}$$ a. \(f(2)\) b. \(f(-2) \quad\) c. \(f(-x)\)

Short Answer

Expert verified
After substituting and simplifying, our answers to the given problems are: a) \(f(2)=\frac{15}{4}\), b) \(f(-2)=\frac{15}{4}\), and c) \(f(-x)= 4 - \frac{1}{x^2}\)

Step by step solution

01

Function substitution

Substitute \(x=2\) into the function \(f(x)\). Thus, \(f(2)=\frac{4 * 2^{2}-1}{2^{2}} = \frac{15}{4}\)
02

Function substitution

Next, substitute \(x=-2\) into the function \(f(x)\). So, \(f(-2)=\frac{4 * (-2)^{2}-1}{(-2)^{2}} = \frac{15}{4}\)
03

Function substitution

Finally, substitute \(x=-x\) into the function \(f(x)\). This will yield \(f(-x)=\frac{4 * (-x)^{2}-1}{(-x)^{2}}\). As \((-x)^2 = x^2\), the equation simplifies to \(f(-x)=\frac{4x^2-1}{x^2} = 4 - \frac{1}{x^2}\)

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Key Concepts

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

Function Substitution
Function substitution is an essential part of evaluating functions. It involves replacing the independent variable with a specific value. In our exercise, we're working with the function \[ f(x)=\frac{4x^2 - 1}{x^2} \]. This requires substituting different values for \(x\).

  • First, for \(f(2)\), you substitute \(x = 2\) into the function. This gives you:\[ f(2) = \frac{4 \times 2^2 - 1}{2^2} = \frac{15}{4} \]
  • Second, for \(f(-2)\), replace \(x\) with \(-2\):\[ f(-2) = \frac{4 \times (-2)^2 - 1}{(-2)^2} = \frac{15}{4} \]
  • Lastly, substituting \(-x\) instead, allows us to see how the function behaves with negative inputs:\[ f(-x) = \frac{4 \times (-x)^2 - 1}{(-x)^2} \]

Recognize that performing substitutions helps in understanding how outputs change relative to inputs, capturing the essence of functions.
Simplification
Simplification is the process of reducing complex expressions into simpler, more manageable forms. It's crucial whenever you evaluate functions to ensure the solution is as clear as possible.

When you substitute a value in a function, simplify what you can:
  • For \(f(2)\) and \(f(-2)\), no further simplification beyond arithmetic was required, as both cases yield \[ \frac{15}{4} \]. This confirms that both substitutions offered the same result.
  • For \(f(-x)\), initial substitution resulted in:\[ \frac{4(-x)^2 - 1}{(-x)^2} \].

    Noticing that \((-x)^2 = x^2\), our function reduces to:\[ \frac{4x^2 - 1}{x^2} \]. Then, you further simplify to:\[ 4 - \frac{1}{x^2} \].

Through simplification, you not only make calculations easier but also unveil more insights into the function's behavior.
Independent Variable
The independent variable in a function is the one you control or input, and it determines the output. For functions like \[ f(x)=\frac{4x^2 - 1}{x^2} \], \(x\) serves as the independent variable.

Understanding the role of the independent variable is essential:
  • Each unique value substituted in for \(x\) results in a distinct output. It's the variety of inputs that allow you to fully comprehend how the function behaves鈥攆or instance, comparing outcomes like \(f(2)\), \(f(-2)\), and \(f(-x)\).
  • The independent variable helps in determining the symmetries and transformations of the function, as seen when \(f(-2)\) yielded the same output as \(f(2)\). This might imply even function characteristics when further examined with other values.

Through exploring different values, you uncover the depths of the function's behavior, leveraging the independent variable's powers as a tool for exploration.

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Most popular questions from this chapter

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. $$ T(40,000) $$

Which one of the following is true? a. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of three units to the right and three units upward of the graph of \(f\) b. If \(f(x)=-\sqrt{x}\) and \(g(x)=\sqrt{-x},\) then \(f\) and \(g\) have identical graphs. c. If \(f(x)=x^{2}\) and \(g(x)=5\left(x^{2}-2\right),\) then the graph of \(g\) can be obtained from the graph of \(f\) by stretching \(f\) five units followed by a downward shift of two units. d. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting in the \(x\) -axis, and then moving the resulting graph down four units.

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=(x-1)^{2} $$

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-2 $$
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