Problem 48 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

(a) \( \sqrt{5} \), (b) \( \sqrt{2 - b} \), (c) \( \sqrt{2 - x^3} \), (d) \( \sqrt{5 - 2x} \)."

Step 1: Evaluate g(-3)

Given that \( g(x) = \sqrt{2-x} \), we need to evaluate \( g(-3) \). Substitute \( -3 \) for \( x \) in the function: \( g(-3) = \sqrt{2 - (-3)} = \sqrt{2 + 3} = \sqrt{5} \). Thus, \( g(-3) = \sqrt{5} \).

Step 2: Evaluate g(b)

Substitute \( b \) for \( x \) in the function \( g(x) \): \( g(b) = \sqrt{2 - b} \). As there is no simplification possible, \( g(b) = \sqrt{2 - b} \).

Step 3: Evaluate g(x^3)

Substitute \( x^3 \) for \( x \) in the function \( g(x) \): \( g(x^3) = \sqrt{2 - x^3} \). Since there's no further simplification, \( g(x^3) = \sqrt{2 - x^3} \).

Step 4: Evaluate g(2x-3)

Substitute \( 2x-3 \) for \( x \) in the function \( g(x) \): \( g(2x-3) = \sqrt{2 - (2x-3)} = \sqrt{2 - 2x + 3} = \sqrt{5 - 2x} \). Thus, \( g(2x-3) = \sqrt{5 - 2x} \).

Key Concepts

Function Evaluation

Function evaluation involves determining the result of a function when a specific value is placed for its variable. In the exercise above, given the function \( g(x) = \sqrt{2-x} \), we are tasked with finding the value of the function when different expressions replace \( x \), such as \(-3\), \(b\), \(x^3\), and \(2x-3\).
To evaluate a function at a particular value, follow these steps:

• Replace the variable in the function with the given value or expression.
• Simplify the expression if possible.

For example, to find \( g(-3) \), we substitute \(-3\) into the function resulting in \( \sqrt{5} \). Function evaluation is a fundamental concept in algebra as it forms the basis of understanding how changes in input affect the output.

Function Substitution

Substitution in functions refers to replacing the variable inside a function with a given number or expression. This task helps to determine how the output changes with different inputs. In our specific example of \( g(x) = \sqrt{2-x} \), we are substituting \(-3\), \(b\), \(x^3\), and \(2x-3\) into the function.
Let's break down a few substitution examples:

• To substitute \( b \) into the function: replace \( x \) with \( b \), resulting in \( g(b) = \sqrt{2 - b} \).
• For \( x^3 \), substitute directly to obtain \( g(x^3) = \sqrt{2 - x^3} \).

The process of substitution is crucial for evaluating functions over a range of different inputs, enabling the calculation of multiple function outputs without redefining the entire expression each time.

Radical Expressions

Radical expressions involve roots, such as square roots, cubic roots, etc. In our example function \( g(x) = \sqrt{2-x} \), the square root sign \( \sqrt{} \) represents the radical expression. Radical expressions can sometimes be intimidating due to their unique properties.
When working with radicals, it's important to ensure the expression under the radical sign is non-negative if dealing with real numbers, as we are here. Here are some basic reminders when handling radicals:

• Simplify expressions under the radical whenever possible.
• For expressions like \( \sqrt{2-x} \), the term \(2-x\) needs to be positive to have a real number output.

Radicals can frequently appear in various math problems. Understanding them will simplify evaluating and manipulating expressions involving roots.

Step-by-Step Solution

Step-by-step solutions are very helpful for breaking down complex problems into individual, manageable parts. This approach is used in the solution to evaluate \( g(x) \) at different expressions in the given problem.
In the provided solution:

• For \( g(-3) \), each step clearly shows substituting \(-3\) and simplifying to \( \sqrt{5} \).
• When replacing with expressions like \( 2x - 3 \), it demonstrates rearranging terms to get \( \sqrt{5 - 2x} \).

With a step-by-step approach, each operation and each substitution is explained clearly. This method ensures understanding and allows identifying where mistakes may occur in function evaluation, substitution, or radical simplification. By following these steps, you can confidently approach similar problems in algebra and get them right.