Problem 48 - 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 values of the function g at the given independent variables are: \(g(0) = 5\), \(g(-6) = 1\), \(g(-5) = 0\).

Step 1: Evaluate g(0)

To evaluate g(0), verify which condition 0 meets in the function. Here, \(x = 0\) is greater than -5, so the first equation applies. Substitute 0 into the first equation: \(0 + 5 = 5\). So, \(g(0) = 5\).

Step 2: Evaluate g(-6)

When evaluating g(-6), \(x = -6\) is less than -5, so the second equation applies. Substitute -6 into the second equation: \(-(-6 + 5) = 1\). So, \(g(-6) = 1\).

Step 3: Evaluate g(-5)

To evaluate g(-5), \(x = -5\) is equal to -5, so the first equation applies. Substitute -5 into the first equation: \(-5+5 = 0\). So, \(g(-5) = 0\).

Key Concepts

Function Evaluation

To understand how to evaluate functions, especially piecewise functions, it's crucial to know which part of the function to use for a given input value. A piecewise function consists of multiple expressions, each corresponding to a different interval of the independent variable (often labeled \( x \)).
When evaluating a piecewise function:

• Identify the correct expression by checking which condition the given \( x \) value satisfies.
• Substitute the \( x \) value into that expression.
• Calculate the result to find the function value.

For example, in the provided exercise, the piecewise function \( g(x) \) consists of two expressions. By determining whether \( x \) is greater than or equal to \(-5\), we decide which part to use. This selection process is essential for correct function evaluation.

Conditional Expressions

Conditional expressions are vital in piecewise functions as they determine which expression to use based on the value of \( x \). In mathematics, these conditions act like filters, guiding which function expression you apply.
Here's how they work in the provided exercise:

• If \( x \geq -5 \), use the expression \( x + 5 \) to calculate the function value.
• If \( x < -5 \), use the expression \(- (x + 5) \) for evaluation.

Ensuring the right expression is picked based on the input value is essential. Each piece of the function has its domain, and observing where your input falls is crucial. This ensures you correctly implement the piece of the function relevant to your \( x \) value. Mastery of conditional expressions helps in dealing with more complex mathematical scenarios.

Algebraic Equations

Solving algebraic equations is a core part of evaluating piecewise functions. Once you've selected the correct piece of the function using the conditions, you'll use algebra to find the answer.
Here’s a quick guide to manage algebraic equations in piecewise functions:

• Simplify the equation by performing basic arithmetic operations such as addition or subtraction.
• Follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Always double-check your substitutions and calculations to ensure accuracy.

Take \( g(-6) \) from this exercise as an example. You apply the equation \( - (x + 5) \) since \(-6\) is less than \(-5\). Calculate with care, ensuring corrections at each step: substitute \(-6\) correctly, perform arithmetic to confirm \( g(-6) = 1 \). By approaching algebraic equations with careful methodology, solving piecewise functions becomes much clearer.