Problem 51 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

(a) Undefined, (b) \(\frac{4b}{b+3}\), (c) \(\frac{4x^3}{x^3+3}\), (d) \(4 - \frac{12}{2x}\)

Step 1: Evaluate \( g(-3) \)

Substitute \( x = -3 \) into the function \( g(x) = \frac{4x}{x+3} \). This gives us \( g(-3) = \frac{4(-3)}{-3+3} = \frac{-12}{0} \). This expression is undefined because division by zero is not allowed.

Step 2: Evaluate \( g(b) \)

Substitute \( x = b \) into the function to get \( g(b) = \frac{4b}{b+3} \). Since \( b \) is not specified, this is our final expression for \( g(b) \), except when \( b = -3 \) where it becomes undefined.

Step 3: Evaluate \( g(x^3) \)

Substitute \( x^3 \) in place of \( x \) in the function: \( g(x^3) = \frac{4(x^3)}{x^3+3} \). This expression will only be undefined if \( x^3 = -3 \), which occurs when \( x = -\sqrt[3]{3} \).

Step 4: Evaluate \( g(2x-3) \)

Substitute \( 2x-3 \) for \( x \) in \( g(x) \): \( g(2x-3) = \frac{4(2x-3)}{(2x-3)+3} = \frac{8x-12}{2x} \). Simplify this expression to \( g(2x-3) = 4 - \frac{12}{2x} \), except when \( 2x = 0 \) or \( x = 0 \), making it undefined.

Key Concepts

Rational Functions

Rational functions are a key part of algebra and calculus. These functions are defined as the ratio of two polynomials. In simpler terms, a rational function is written as \( g(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
Understanding rational functions involves considering both their numerator and denominator, as they work together to determine the behavior and properties of the function.
The function given in this exercise is \( g(x) = \frac{4x}{x+3} \). Here, \( 4x \) is the numerator and \( x+3 \) is the denominator. Rational functions can demonstrate interesting behaviors and can sometimes become undefined, which we will explore further.

Division by Zero

Division by zero is an important concept in mathematics, often associated with undefined expressions. The principle is simple: dividing any number by zero doesn't yield a valid result.
When we attempt to calculate an expression where the denominator equals to zero, it becomes undefined. This is because dividing by zero disrupts the basic properties of arithmetic.
In the context of rational functions, checking the denominator for values that make it zero is crucial. If \( Q(x) = 0 \), the function is not valid for that input, and it's considered undefined. For \( g(x) = \frac{4x}{x+3} \), division by zero occurs at \( x = -3 \). This is why, when evaluating \( g(-3) \), we find that the function is undefined.

Input Substitution

Substitution is a core concept in function evaluation. It involves replacing the independent variable in the function with a specific value or expression to find the output.
For instance, to evaluate \( g(b) \), we substitute \( x \) with \( b \) in \( g(x) = \frac{4x}{x+3} \). This gives \( g(b) = \frac{4b}{b+3} \).
Another example from the exercise is \( g(x^3) \), where we replace \( x \) with \( x^3 \), leading to \( g(x^3) = \frac{4(x^3)}{x^3+3} \). Input substitution allows us to explore the behavior of functions for different inputs, providing a versatile tool for analyzing mathematical scenarios. This concept requires attention to ensure no division by zero occurs in the new expression, as noted when \( x^3 = -3 \).

Undefined Expressions

Undefined expressions often arise in mathematics when certain operations cannot be completed under conventional arithmetic rules, like division by zero. Identifying when a function becomes undefined is crucial for understanding its domain and range.
For the function \( g(x) = \frac{4x}{x+3} \), the expression is undefined at \( x = -3 \), as earlier mentioned. This is because the denominator \( x + 3 = 0 \) creates division by zero.
In a more complex scenario, such as \( g(2x-3) = \frac{8x-12}{2x} \), simplifying to \( 4 - \frac{12}{2x} \), the expression is undefined if \( 2x = 0 \) specifically, \( x = 0 \). Understanding where a function is undefined helps in sketching its graph and clarifying its properties, such as finding asymptotes and points of discontinuity.