Problem 47 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

(a) 1; (b) \( \sqrt{b+4} \); (c) \( \sqrt{x^3+4} \); (d) \( \sqrt{2x+1} \)."

Step 1: Evaluate g(-3)

To find \( g(-3) \), substitute \(-3\) into the function \( g(x) = \sqrt{x+4} \). Calculate \( g(-3) = \sqrt{-3+4} = \sqrt{1} = 1 \).

Step 2: Evaluate g(b)

To find \( g(b) \), substitute \( b \) into the function \( g(x) = \sqrt{x+4} \). This gives \( g(b) = \sqrt{b+4} \). Since this expression does not simplify without a specific value for \( b \), it remains \( \sqrt{b+4} \).

Step 3: Evaluate g(x^3)

With the function \( g(x) = \sqrt{x+4} \), substitute \( x^3 \) in place of \( x \) to get \( g(x^3) = \sqrt{x^3 + 4} \). This result is \( \sqrt{x^3+4} \) since it can't be simplified further without a specific value for \( x \).

Step 4: Evaluate g(2x-3)

Substitute \( 2x-3 \) into \( g(x) = \sqrt{x+4} \). Calculate \( g(2x-3) = \sqrt{(2x-3)+4} = \sqrt{2x+1} \). This result is \( \sqrt{2x+1} \).

Key Concepts

Square Root Function

The square root function is a type of radical function that is central to many mathematical computations. When you see an expression like \( g(x) = \sqrt{x+4} \), you're looking at a simple square root function. The expression under the square root, in this case \( x+4 \), is known as the radicand. The square root function itself involves finding a value, which when multiplied by itself, returns the radicand.The domain of a standard square root function is all non-negative numbers since the square root of a negative number is not defined in the set of real numbers. For \( g(x) = \sqrt{x+4} \), for example, the domain of \( x \) would be values such that \( x+4 \geq 0 \), or \( x \geq -4 \). The range, meanwhile, is all non-negative numbers because a square root always produces non-negative results.

Substitution in Functions

Substitution in functions is a critical process in algebra which involves replacing the variable in a function with a specific value or another expression. This method allows us to evaluate the function for different inputs without modifying the original function. Consider the function \( g(x) = \sqrt{x+4} \).- To evaluate \( g(-3) \), you replace \( x \) with \(-3\) leading to \( g(-3) = \sqrt{-3+4} = 1 \).- For \( g(b) \), substitution is straightforward: \( b \) is just placed in for \( x \), giving \( g(b) = \sqrt{b+4} \).Substitution can also include more complex expressions, like \( x^3 \) or \( 2x-3 \), showing how a function behaves for various types of inputs.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations that are combined according to specific algebraic rules. Understanding these expressions is crucial when working with functions. With interfacing functions like \( g(x) = \sqrt{x+4} \), you can have different variations where algebraic expressions are substituted for the variable.Take \( g(x^3) \), where the expression \( x^3 \) replaces \( x \). This results in \( g(x^3) = \sqrt{x^3+4} \). In this context, \( x^3+4 \) is an algebraic expression within the radicand of a square root function.Evaluating these types of expressions might not simplify neatly or result in a single number but understanding them allows us to manipulate complex functions and solve equations systematically.

Function Transformations

Function transformations involve altering a function to produce a new function with certain desired properties. These changes can shift, stretch, or compress the graph of the original function. In the context of \( g(x) = \sqrt{x+4} \), evaluating \( g(2x-3) \) involves such transformations.Substituting \( 2x-3 \) into the function yields \( g(2x-3) = \sqrt{2x-3+4} = \sqrt{2x+1} \). This transformation includes horizontal shifts and stretches:- The term \( 2x \) represents a horizontal stretch by a factor of 2.- The \(-3\) and \(+4\) adjust the function by first shifting it left and then right, respectively.Understanding function transformations helps in visualizing how inputs to a function produce results, allowing us to predict how changes to input variables affect the function's graph and outputs.