Problem 49 - College Algebra with Modeling and Visualization

Question

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=|3 x-1|+4 $$

Step-by-Step Solution

Answer

(a) 14, (b) |3b - 1| + 4, (c) |3x^3 - 1| + 4, (d) |6x - 10| + 4.

Step 1: Substituting -3 into g(x)

To find \( g(-3) \), substitute \( x = -3 \) into the function. We have:\[ g(-3) = |3(-3) - 1| + 4 \] Compute inside the absolute value first: \[ 3(-3) - 1 = -9 - 1 = -10 \] Take the absolute value: \[ |-10| = 10 \] Add 4:\[ g(-3) = 10 + 4 = 14 \]

Step 2: Substituting b into g(x)

To find \( g(b) \), substitute \( x = b \) into the function. We have:\[ g(b) = |3b - 1| + 4 \] This is the expression for \( g(b) \); simplify further if possible given a value for \( b \).

Step 3: Substituting x³ into g(x)

To evaluate \( g(x^3) \), substitute \( x^3 \) for \( x \) in the function:\[ g(x^3) = |3(x^3) - 1| + 4 \] Simplifying, we find:\[ g(x^3) = |3x^3 - 1| + 4 \]This is the expression for \( g(x^3) \).

Step 4: Substituting 2x - 3 into g(x)

To evaluate \( g(2x - 3) \), substitute \( 2x - 3 \) for \( x \) in the function:\[ g(2x - 3) = |3(2x - 3) - 1| + 4 \] Compute inside the absolute value: \[ 3(2x - 3) = 6x - 9 \] So: \[ 6x - 9 - 1 = 6x - 10 \] Thus:\[ g(2x - 3) = |6x - 10| + 4 \]This is the expression for \( g(2x - 3) \).

Key Concepts

Absolute Value Functions

An absolute value function, like \( g(x) = |3x - 1| + 4 \), is essentially made up of absolute values. The absolute value, denoted by the vertical bars \(| \ |\), captures how far a number is from zero on the number line.
Regardless if the inside value is negative or positive, the absolute value converts it to a non-negative number.To evaluate an absolute value function, always:

• Compute the expression inside the absolute value first.
• Take the absolute value of the obtained result.
• Proceed with any remaining operations, like addition.

For example, calculating \( g(-3) \) requires you to deal with the expression inside \(| \ |\) first: substitute \( -3 \), find \(-10\), take its absolute value to get \(10\), and then solve \(10 + 4\) to find the value of \( g(-3) \) as \(14\).
This method ensures you handle all parts of the function correctly.

Substitution in Functions

Function evaluation often involves substituting a given value or expression into the function. This means replacing the variable within the function definition with the specified value or expression.
For example, if you have a function \( g(x) = |3x - 1| + 4 \), then \( g(b) \) means you substitute \( b \) for \( x \):

• First, write the function: \( g(x) = |3x - 1| + 4 \)
• Then, substitute \( b \) for \( x \): \( g(b) = |3b - 1| + 4 \)

The function remains in terms of \( b \) until a numerical value is provided to simplify further.
This same substitution technique applies regardless of whether you're substituting numbers, variables, or even more complex expressions like \( x^3 \).
Substitution is a fundamental skill in mathematics, allowing evaluation and manipulation of function expressions.

Piecewise Function Evaluation

Piecewise functions can have different rules applied over different intervals of the input variable. In the context of this exercise, although \( g(x) = |3x - 1| + 4 \) itself isn't truly a piecewise function, understanding piecewise evaluation helps us handle similar function evaluations tactically.
Since absolute value functions sometimes result in piecewise representations, here's how to think about evaluating such functions:

• Identify the intervals where each piece applies.
• Apply the correct function rule based on the input value's position within these intervals.
• Evaluate the expression using the appropriate operation for the specific interval.

While \( g(x) \) isn't split into pieces explicitly, knowing how operations change with different conditions (positive or negative input) can aid in function evaluation. Hence, each evaluation is contextual and specific to input values, ensuring the correct evaluation approach is taken for any piecewise-related tasks.