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Chapter 3: Problem 2

Consider the following function: $$ g(x)=\left\\{\begin{array}{ll} 3 & x \leq 1 \\ 4+2 x & x>1 \end{array}\right. $$ Evaluate \(g(-5), g(-2), g(0), g(1), g(1.1), g(2),\) and \(g(10)\)

Short Answer

Expert verified
g(-5) = 3, g(-2) = 3, g(0) = 3, g(1) = 3, g(1.1) = 6.2, g(2) = 8, g(10) = 24.

Step by step solution

01

Evaluate \(g(x)\) for \(x \leq 1\)

The function definition tells that for any \(x\) less than or equal to 1, \(g(x) = 3\). Therefore, for \(x = -5\), \(x = -2\), \(x = 0\), and \(x = 1\): \[ g(-5) = 3 \]\[ g(-2) = 3 \]\[ g(0) = 3 \]\[ g(1) = 3 \]
02

Evaluate \(g(x)\) for \(x > 1\)

For any \(x\) greater than 1, the second part of the function definition applies, where \(g(x) = 4 + 2x\). Therefore, for \(x = 1.1\), \(x = 2\), and \(x = 10\):\[ g(1.1) = 4 + 2(1.1) = 4 + 2.2 = 6.2 \]\[ g(2) = 4 + 2(2) = 4 + 4 = 8 \]\[ g(10) = 4 + 2(10) = 4 + 20 = 24 \]

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Key Concepts

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

function evaluation
To evaluate a function means to find the function's value at a specific input. This involves substituting the input value into the function's equation and performing the necessary arithmetic operations to determine the result.

The function we are working with is a piecewise function, so depending on the input value, different rules may be used for the calculation. By substituting each input value into the correct part of the piecewise function, we can find the corresponding output values.
piecewise-defined functions
A piecewise-defined function is one that has different expressions based on the input value. It may use one rule for some values of the input and a different rule for others. This allows the function to behave differently over different regions of its domain.

For instance, consider the function provided:
\ \[ g(x)=\left\{\begin{array}{ll} 3 & x \leq 1 \ 4+2 x & x>1 \end{array}\right. \]
This function has two distinct parts:
  • For any input value \(x \leq 1\), \(g(x)\) will always be 3.
  • For any input value \(x > 1\), \(g(x)\) will be calculated using the rule \(4 + 2x\).
Piecewise functions are useful for modeling scenarios where different rules apply to different situations, providing flexibility in representing complex behaviors.
algebraic expressions
Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\)), and operations (like addition or multiplication).

Consider the expression \(4 + 2x\) in our piecewise function. This expression consists of:
  • A constant term (4)
  • A variable term (\(2x\)), which means 2 times the input value
To evaluate an algebraic expression, we substitute the given value of the variable and perform the arithmetic operations. For example, if \(x = 2\) in the expression \(4 + 2x\), we calculate: \ \[ 4 + 2(2) = 4 + 4 = 8 \]
This process is essential to understanding how different parts of a piecewise function work.
step-by-step solutions
Solving problems step-by-step is a reliable method for understanding and verifying solutions. Here's how we apply it to evaluate the given piecewise function:

Step 1: Identify which part of the function to use. The primary step is to check if the input value is \( \leq 1 \) or \( > 1 \). Depending on this, we use the respective part of the piecewise function.

Step 2: Substitute the input value. For \(x = -5, -2, 0, 1\), since all these numbers are \( \leq 1 \), we use \( g(x) = 3 \). Thus:
  • \( g(-5) = 3 \)
  • \( g(-2) = 3 \)
  • \( g(0) = 3 \)
  • \( g(1) = 3 \)


For \(x = 1.1, 2, 10\), since these numbers are \(> 1\), we use \( g(x) = 4 + 2x \). Thus:
  • \( g(1.1) = 4 + 2(1.1) = 6.2 \)
  • \( g(2) = 4 + 2(2) = 8 \)
  • \( g(10) = 4 + 2(10) = 24 \)
Following these systematic steps helps avoid mistakes and provides a clear path to the right answer.

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Most popular questions from this chapter

Calculate the solution(s), if any, to each of the following systems of equations. Use any method you like. $$ \begin{array}{ll} \text { a. } y=-1-2 x & \text { c. } y=2200 x-700 \\ y=13-2 x & y=1300 x+4700 \\ \text { b. } t=-3+4 w & \text { d. } 3 x=5 y \\ -12 w+3 t+9=0 & 4 y-3 x=-3 \end{array} $$

a. Determine whether (5,-10) is a solution for the following system of equations: $$ \begin{array}{l} 4 x-3 y=50 \\ 2 x+2 y=5 \end{array} $$ b. Explain why (-10,5) is not a solution for the system in \(\operatorname{part}(a)\)

(Graphing program required.) Two professors from Purdue University reported that for a typical small-sized fertilizer plant in Indiana the fixed costs were \(\$ 235,487\) and it cost \(\$ 206.68\) to produce each ton of fertilizer. a. If the company planned to sell the fertilizer at \(\$ 266.67\) per ton, find the cost, \(C,\) and revenue, \(R\), equations for \(x\) tons of fertilizer. b. Graph the cost and revenue equations on the same graph and calculate and interpret the breakeven point. c. Indicate the region where the company would make a profit and create the inequality to describe the profit region.

Construct a small table of values and graph the following piecewise linear functions. In each case specify the domain. a. \(f(x)=\left\\{\begin{array}{ll}5 & \text { for } x<10 \\ -15+2 x & \text { for } x \geq 10\end{array}\right.\) b. \(g(t)=\left\\{\begin{array}{ll}1-t & \text { for }-10 \leq t \leq 1 \\ t & \text { for } 1

A husband drives a heavily loaded truck that can go only 55 mph on a 650 -mile turnpike trip. His wife leaves on the same trip 2 hours later in the family car averaging \(70 \mathrm{mph}\). Recall that distance traveled \(=\) speed \(\cdot\) time traveled. a. Derive an expression for the distance, \(D_{h},\) the husband travels in \(t\) hours since he started. b. How many hours has the wife been traveling if the husband has traveled \(t\) hours \((t \geq 2) ?\) c. Derive an expression for the distance, \(D_{w}\), that the wife will have traveled while the husband has been traveling for \(t\) hours \((t \geq 2)\) d. Graph distance vs. time for husband and wife on the same axes. e. Calculate when and where the wife will overtake the husband. f. Suppose the husband and wife wanted to arrive at a restaurant at the same time, and the restaurant is 325 miles from home. How much later should she leave, assuming he still travels at \(55 \mathrm{mph}\) and she at \(70 \mathrm{mph} ?\)
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