Problem 40 - College Algebra

Question

Evaluate each piecewise function at the given values of the independent variable. $$g(x)=\left\\{\begin{array}{ll}x+5 & \text { if } x \geq-5 \\\\-(x+5) & \text { if } x<-5\end{array}\right.$$ $$a. g(0)$$ $$b. g(-6)$$ $$c. g(-5)$$

Step-by-Step Solution

Answer

The evaluated functions are as follows: g(0) = 5, g(-6) = 1 and g(-5) = 0.

Step 1: Evaluate g(0)

For this, it is necessary to identify which part of the function applies to x = 0. As 0 ≥ -5, the first piece of the function is applicable. So, substitute x = 0 into the equation of the first piece, which becomes \(0 + 5 = 5\). Therefore, g(0) = 5.

Step 2: Evaluate g(-6)

For this, it is necessary to identify which part of the function applies to x = -6. As -6 < -5, the second piece of the function is applicable. So, substitute x = -6 into the equation of the second piece, which becomes \(-(-6 + 5) = -(-1) = 1\). Therefore, g(-6) = 1.

Step 3: Evaluate g(-5)

For this, it is necessary to identify which part of the function applies to x = -5. As -5 is equal to -5, the first piece of function applies. So, substitute x = -5 into the equation of the first piece, which becomes \(-5 + 5 = 0\). Therefore, g(-5) = 0.

Key Concepts

Function Evaluation

Function evaluation deals with finding the output of a function for a given input. In the case of piecewise functions, which are made up of multiple "pieces" or parts, it's important to evaluate each input by determining which part it falls under.

With the function given here, there are two expressions that depend on the input value (x).Each piece of the function comes with a specific rule:

• Use the first expression, \(x + 5\), if \(x \geq -5\).
• Use the second expression, \(-(x + 5)\), if \(x < -5\).

Whenever you evaluate a piecewise function, always check which rule applies to your input before performing the calculation. It’s straightforward once you match the condition and then apply the corresponding mathematical operation. This method simplifies the process of calculating outputs accurately based on varying conditions.

Independent Variable

In mathematics, the independent variable is the variable that you can freely change or control. Often, it's labeled as \(x\). It's the input that you plug into your function to test different scenarios or calculations.

For piecewise functions like \(g(x)\), the value of \(x\) determines which of the different function pieces will be applied. This is crucial to understanding function behavior because each segment might have different mathematical rules.

• If you choose \(x = 0\), the independent variable makes \(g(x)\) calculate using one rule.
• However, choosing \(x = -6\) leads to using another part of the function.

In essence, understanding the independent variable lets you predict how the function will respond to varying inputs. It is an essential part of analyzing piecewise functions and making sure evaluations are correct.

Conditional Expressions

Conditional expressions in piecewise functions determine which formula to apply from the function's segments based on the value of the independent variable. Each piece has a specific condition that decides if it should be used.
For the function \(g(x)\), we see two conditional expressions:

• The first applies when \(x \geq -5\).
• The second applies when \(x < -5\).

Evaluating \(g(0)\) or \(g(-6)\) relies on these conditions to guide you to the correct calculation. If you're examining \(g(0)\), recognize that \(0 \geq -5\), hence you use the first expression \(x + 5\), making the output 5.

For \(g(-6)\), since \(-6 < -5\), you leverage the second expression \(-(x + 5)\) and substitute \(x\) resulting in an outcome of 1. Conditional expressions act as decision-makers, streamlining which mathematical operation to use and ensuring function accuracy.