Problem 50 - College Algebra with Modeling and Visualization

Question

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2|1-x|-7 $$

Step-by-Step Solution

Answer

(a) 1, (b) \(2|1-b|-7\), (c) \(2|1-x^3|-7\), (d) \(2|4-2x|-7\)

Step 1: Evaluate g(-3)

To evaluate \( g(-3) \), substitute \( -3 \) for \( x \) in the function \( g(x) = 2|1-x|-7 \).\[g(-3) = 2|1-(-3)|-7 = 2|1+3|-7 = 2|4|-7 = 8-7 = 1\]Thus, \( g(-3) = 1 \).

Step 2: Substitute and Simplify g(b)

To find \( g(b) \), replace \( x \) with \( b \) in the function:\[g(b) = 2|1-b|-7\]This expression is already simplified, so \( g(b) = 2|1-b|-7 \).

Step 3: Evaluate g(x^3)

Substitute \( x^3 \) for \( x \) in the function:\[g(x^3) = 2|1-x^3|-7\]This expression is already simplified, so \( g(x^3) = 2|1-x^3|-7 \).

Step 4: Evaluate g(2x-3)

Substitute \( 2x-3 \) for \( x \) in the function:\[g(2x-3) = 2|1-(2x-3)|-7 = 2|1-2x+3|-7 = 2|4-2x|-7\]Thus, \( g(2x-3) = 2|4-2x|-7 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra that involves finding the output of a function for a given input. Imagine a function like a machine: you input a value, and it processes it to give you an output. To evaluate a function like \( g(x) = 2|1-x|-7 \), we replace the variable \( x \) with any specified value or expression.

For example, when evaluating \( g(-3) \), we substitute \( -3 \) in place of \( x \), which leads to performing operations within the function to find the output. It's important to follow arithmetic and algebraic rules carefully during substitution and simplification.

• Substitute the specified value or expression.
• Simplify the expression step-by-step.
• Apply any function-specific rules, such as absolute value, if present.

As demonstrated, function evaluation is a step-by-step process that enables mathematicians to determine the value of a function at any point. It's a critical skill in solving many algebraic problems efficiently.

Absolute Value Functions

Absolute value functions are a special type of function that represent the distance of a number from zero on the number line. The absolute value symbol is denoted by vertical bars, like \( |x| \).Absolute values ensure that any negative input becomes positive, as distance cannot be negative. For instance, \( |1-(-3)| = |1+3| = |4| = 4 \). In the function \( g(x) = 2|1-x|-7 \), the presence of \(|1-x| \) means any outcome from \( (1-x) \) is converted to a non-negative number before continuing with the function's operations.

Key points to remember:

• Absolute values turn all outputs to positive unless they are already positive or zero.
• They influence the operation order by being resolved before multiplication, addition, or subtraction in expressions.
• They can impact the shape of graphs, often resulting in V-shaped curves.

Using absolute values in functions requires careful attention to the sign of numbers within the bars, as it can significantly affect the outcome.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operation symbols like addition or multiplication. They are more than just simple equations; they represent a set of calculations to be performed. Evaluating algebraic expressions involves simplifying them by following mathematical rules and operations.

When handling an expression like \( 2|1-x|-7 \), we need to apply operations in the correct order. The function \( g(x) = 2|1-x|-7 \) features multiplication and absolute value, requiring proper management of both to simplify the expression correctly.

• Work through operations inside grouping symbols like parentheses or absolute value symbols first.
• Perform any multiplication or division from left to right.
• Finally, tackle addition and subtraction.

These steps ensure that you simplify the expression following standard operational hierarchy, avoiding calculation mistakes. Mastering how to manipulate algebraic expressions is crucial for solving complex equations and understanding algebra deeply.