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Chapter 2: Problem 7

For each piecewise-defined function, find (a) \(f(-5),\) (b) \(f(-1)\) (c) \(f(0),\) and (d) \(f(3) .\) Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} 2+x & \text { if } x<-4 \\ -x & \text { if }-4 \leq x \leq 2 \\ 3 x & \text { if } x>2 \end{array}\right.$$

Short Answer

Expert verified
\( f(-5) = -3, \; f(-1) = 1, \; f(0) = 0, \; f(3) = 9. \)

Step by step solution

01

Understanding the Piecewise Function

The function \( f(x) \) is defined by different expressions depending on the value of \( x \). For \( x < -4 \), it uses \( f(x) = 2 + x \). For \( -4 \leq x \leq 2 \), it uses \( f(x) = -x \). For \( x > 2 \), it uses \( f(x) = 3x \). We will use these expressions to find \( f(-5) \), \( f(-1) \), \( f(0) \), and \( f(3) \).
02

Evaluating f(-5)

Since \( -5 < -4 \), use the expression \( f(x) = 2 + x \). Substitute \( x = -5 \) to get \( f(-5) = 2 + (-5) = -3 \).
03

Evaluating f(-1)

Since \( -4 \leq -1 \leq 2 \), use the expression \( f(x) = -x \). Substitute \( x = -1 \) to get \( f(-1) = -(-1) = 1 \).
04

Evaluating f(0)

Since \( -4 \leq 0 \leq 2 \), use the expression \( f(x) = -x \). Substitute \( x = 0 \) to get \( f(0) = -(0) = 0 \).
05

Evaluating f(3)

Since \( 3 > 2 \), use the expression \( f(x) = 3x \). Substitute \( x = 3 \) to get \( f(3) = 3(3) = 9 \).

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Key Concepts

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

Evaluating Functions
Evaluating functions is the process of finding the value of a function at a specific point. It involves plugging in a given value for the input variable (usually represented as "x") into the function. This allows us to determine what the function outputs at that particular input.

For piecewise functions, evaluating the function depends on which part of the piecewise formula applies to the input value. This means not every piece of the function is used for each evaluation. Instead, the correct expression is chosen based on the input value's value.

When evaluating:
  • Identify the condition that the input value satisfies (e.g. "x < -4", "-4 鈮� x 鈮� 2", "x > 2").
  • Substitute the input value into the appropriate expression under that condition.
  • Compute the result to find the function's value at the input.
Through this step-by-step approach, evaluating a function becomes clear and manageable.
Functions
Functions are mathematical expressions that relate inputs (often "x") to outputs. In other words, a function is like a machine that takes each input, processes it according to a rule (or set of rules), and provides an output.

Functions can be represented in various ways, such as equations, graphs, or tables. Each representation gives a different perspective on how the function behaves. For example, on a graph, you can visually see the slope and intercept points, while a table allows you to see discrete input-output pairs.

It's important to understand:
  • Each input in a function can have only one output.
  • The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs.
  • Piecewise functions are special because they use different rules depending on which input is being evaluated.
Functions are foundational concepts in algebra and calculus and are widely used in various fields such as engineering and economics.
Expression Substitution
Expression substitution involves replacing a variable in an expression with a specific value or another expression. This process is key when working with functions, especially piecewise functions, as it allows us to evaluate and simplify them.

In these cases:
  • Identify which part of the function or expression is relevant for the given condition.
  • Substitute the value or another expression in place of the variable.
  • Simplify the resulting expression to find your answer.
This method is useful not only in evaluating functions but also in solving equations and re-arranging formulas. It ensures precise results by systematically replacing and reducing expressions. By mastering expression substitution, students can solve complex problems with confidence.

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Most popular questions from this chapter

Solve each problem. When a model kite was flown in crosswinds in tests, it attained speeds of 98 to 148 feet per second in winds of 16 to 26 feet per second. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=3-6 x$$

Consider the function \(h\) as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\). (There are several possible ways to do this.) $$h(x)=\sqrt{6 x}+12$$

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=4 x-8\) must be less than 0.04 unit from 4.

Video-on-Demand The following table shows the projected revenue earned in various years by the U.S. "Video-On-Demand" market segment in millions of dollars. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Revenue (in 5 millions) } \\ \hline 2015 & 9040 \\ 2016 & 9529 \\ 2017 & 10,000 \\ 2018 & 10,436 \\ 2019 & 10,825 \\ 2020 & 11,162 \\ 2021 & 11,448 \\ \hline \end{array}$$ (a) Use a calculator to find the least-squares regression line for these data, where \(x\) is the number of years after 2015 (b) Based on your result from part (a), write an equation that yields the same \(y\) -values when the actual year is entered. (c) Predict the revenue for this market segment to the nearest million dollars in 2025 .
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