Problem 44 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

(a) -16, (b) \(-(b-1)^2\), (c) \(-((x^3)-1)^2\), (d) \(-(2x-4)^2\)."

Step 1: Evaluate g(-3)

To find \( g(-3) \), substitute \( x = -3 \) into the function \( g(x) = -(x-1)^2 \). Calculate as follows:\[g(-3) = -((-3) - 1)^2 = -(-4)^2 = -16\]Thus, \( g(-3) = -16 \).

Step 2: Evaluate g(b)

Substitute \( x = b \) directly into the function \( g(x) = -(x-1)^2 \):\[g(b) = -(b - 1)^2\]Hence, \( g(b) = -(b - 1)^2 \).

Step 3: Evaluate g(x^3)

Substitute \( x = x^3 \) into \( g(x) \):\[g(x^3) = -((x^3) - 1)^2\]Therefore, \( g(x^3) = -((x^3) - 1)^2 \).

Step 4: Evaluate g(2x-3)

Substitute \( x = 2x - 3 \) into \( g(x) \):\[g(2x-3) = -((2x - 3) - 1)^2 = -(2x - 4)^2\]Thus, \( g(2x-3) = -(2x-4)^2 \).

Key Concepts

Function Notation

Function notation is a mathematical shorthand to define and evaluate functions effortlessly. It uses symbols to represent functions clearly and succinctly. For example, the function \( g(x) = -(x-1)^2 \) describes a relationship between the input \( x \) and its output. The letter \( g \) is often used to denote the function's name, and \( x \) is the variable or the input of the function.

Using function notation makes it easier to plug in different numbers or expressions for the variable \( x \) and get a specific result. It acts like a formula or rule that prescribes how to manipulate \( x \) to find the result or output. When substituting a value into the function, the notation \( g(x) \) helps you quickly understand that the value is being placed in a formula instead of exploring the entire equation each time.

Function notation also highlights how functions can be applied to analyze patterns, control variability, and even solve equations. It provides a structured way to work with mathematical relationships efficiently.

Substitution in Functions

Substitution in functions is a straightforward process and is integral to evaluating and manipulating functions. It involves replacing the function's variable with a given number, expression, or another variable to determine the output.

Here are some tips to keep substitution clear:

• Identify the function equation, like \( g(x) = -(x-1)^2 \) in the examples.
• Determine the value or expression you will substitute for \( x \), such as \(-3\), \(b\), \(x^3\), or \(2x-3\).
• Replace every instance of \( x \) in the function with the chosen value or expression.
• Perform the arithmetic operations carefully to arrive at the final result.

Substitution helps in exploring the behavior of functions under different conditions. It is useful in analyzing how changes in input affect the output and in comparing different function values directly. This method is not only useful for specific calculations but also for building understanding in algebraic manipulations.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 and have the general form \( f(x) = ax^2 + bx + c \). In the exercise, the function \( g(x) = -(x-1)^2 \) is a quadratic function written in a slightly different, yet equivalent, form.

Here's a breakdown of important aspects of quadratic functions:

• The degree of the quadratic function is always 2, which means the highest power of \( x \) is squared.
• Quadratics often create parabolic shapes when graphed, either opening upwards or downwards depending on the sign of the coefficient \( a \).
• In the function \( g(x) = -(x-1)^2 \), expanding it gives \( g(x) = -(x^2 - 2x + 1) \), hence \( a = -1 \), \( b = 2 \), and \( c = -1 \). This shows the function results in a parabola opening downwards because the coefficient of \( x^2 \) is negative.
• Quadratic functions have a vertex, which serves as the highest or lowest point on the graph. For \( g(x) = -(x-1)^2 \), the vertex is at \( (1, 0) \).

Understanding quadratic functions is crucial as they appear frequently in various fields such as physics, engineering, and economics. Their parabolic patterns are used to model situations where there is acceleration or a symmetrical distribution of values.