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Chapter 1: Problem 21

For each function: $$ f(x)=\sqrt{-x} ; \text { find } f(-25) $$

Short Answer

Expert verified
The value of \( f(-25) \) is 5.

Step by step solution

01

Understand the Function

The function provided is \( f(x) = \sqrt{-x} \). This means that for a given value of \( x \), we must first take the negative of \( x \), and then find the square root of that value.
02

Substitute the Given Value into the Function

To find \( f(-25) \), substitute \( -25 \) for \( x \) in the function. This gives:\[ f(-25) = \sqrt{-(-25)} \]
03

Simplify the Expression

Inside the square root, simplifying \(-(-25)\) gives \(25\). Therefore, we need to calculate:\[ \sqrt{25} \]
04

Calculate the Square Root

The square root of \( 25 \) is \( 5 \), since \( 5 \times 5 = 25 \). Substitute back to conclude that:\[ f(-25) = 5 \]

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Key Concepts

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

Function Evaluation
When dealing with functions, evaluation refers to the process of finding function values at certain given points. In the problem provided, we are given the function \( f(x) = \sqrt{-x} \). To solve for \( f(-25) \), we substitute \(-25\) in place of \(x\) in the function. This means we replace \(x\) with \(-25\) in the expression \(\sqrt{-x}\).

This transformation leads us to \(\sqrt{-(-25)}\). Handling functions often requires recognizing patterns within the expression: in this case, realizing that substituting \(-25\) for \(x\) means effectively taking the square root of the positive equivalent of \(x\). That is the crux of evaluating this function.
  • Substitution is a key step in function evaluation.
  • Understand the functional form before starting evaluations.
  • Ensure all substituted values adhere to definitions or transformations during function evaluations.
Negative Numbers
Negative numbers can sometimes be tricky especially when involved in operations such as finding square roots. In the context of this function, \( f(x) = \sqrt{-x} \), encountering negative numbers is quite deliberate. Negative numbers under the square root often require careful handling because in real numbers, you can't directly take the square root of a negative number.

The trick here involves \(-x\) under the square root which cleverly negates the negative value, flipping it into a positive. In the exercise, \( f(-25) \) is transformed to \(\sqrt{-(-25)}\), where the double negation leads to a positive 25 prepared for square rooting.
  • Double negatives result in positive values.
  • Negation can affect the direction or positivity of a number.
  • Always consider the context provided by the function when dealing with negatives.
Simplifying Expressions
Simplifying expressions makes them easier to work with by reducing complexity. In the exercise, once we substituted \(-25\) into the function, we obtained \(\sqrt{-(-25)}\).

The immediate step involves simplifying the expression by computing \(-(-25)\), which results in a positive 25. This simplification sets the stage for an easy calculation of the square root of 25. Simplifying in this manner ensures you efficiently follow through potentially complicated operations.

After this simple transformation, we then proceed to calculate \(\sqrt{25} = 5\).
  • Look for straightforward simplification steps first.
  • Verify each transformation leads to a reduced and correct form.
  • Keep transformations logical and clear at every step.

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Most popular questions from this chapter

GENERAL: Newsletters A newsletter has a maximum audience of 100 subscribers. The publisher estimates that she will lose 1 reader for each dollar she charges. Therefore, if she charges \(x\) dollars, her readership will be \((100-x)\). a. Multiply this readership by \(x\) (the price) to find her total revenue. Multiply out the resulting quadratic function. b. What price should she charge to maximize her revenue? [Hint: Find the value of \(x\) that maximizes this quadratic function.]

BUSINESS: Straight-Line Depreciation Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ( \({ }^{\prime \prime}\) depreciates" \()\) through use. Given the original price of an asset, its useful lifetime, and its scrap value (its value at the end of its useful lifetime), the value of the asset after \(t\) years is given by the formula: $$ \begin{aligned} \text { Value }=(\text { Price })-&\left(\frac{(\text { Price })-(\text { Scrap value })}{(\text { Useful lifetime })}\right) \cdot t \\ & \text { for } 0 \leq t \leq(\text { Useful lifetime }) \end{aligned} $$ a. A newspaper buys a printing press for $$\$ 800,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 60,000$$. Use the formula above Exercise 63 to find a formula for the value \(V\) of the press after \(t\) years, for \(0 \leq t \leq 20\) b. Use your formula to find the value of the press after 10 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,800,000] .\) [Hint: Use \(x\) instead of \(t\).]

ATHLETICS: Muscle Contraction The fundamental equation of muscle contraction is of the form \((w+a)(v+b)=c\), where \(w\) is the weight placed on the muscle, \(v\) is the velocity of contraction of the muscle, and \(a, b\), and \(c\) are constants that depend upon the muscle and the units of measurement. Solve this equation for \(v\) as a function of \(w, a, b\), and \(c\).

A 5 -foot-long board is leaning against a wall so that it meets the wall at a point 4 feet above the floor. What is the slope of the board? [Hint: Draw a picture.]

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\), where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t\), where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t\). and evaluate the function at \(t=10\).
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