Problem 24 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$g(x)=x^{2}-10 x-3$$ a. \(g(-1)\) b. \(g(x+2) \quad\) c. \(g(-x)\)

Step-by-Step Solution

Answer

The simplified expression for \(g(-1)\) is 8. The simplified expression for \(g(x+2)\) is \(x^{2} - 6x - 19\). The simplified expression for \(g(-x)\) is \(x^{2} + 10x - 3\).

Step 1: Substituting and Simplifying for \(g(-1)\)

First, replace \(x\) in the function with -1: \(g(-1) = (-1)^{2} - 10(-1) - 3 = 1 + 10 - 3 = 8\)

Step 2: Substituting and Simplifying for \(g(x+2)\)

Now, substitute \(x+2\) for \(x\) in the function: \(g(x+2) = (x+2)^{2} - 10(x+2) - 3\). This simplifies to: \(x^{2} + 4x + 4 - 10x - 20 - 3 = x^{2} - 6x - 19\)

Step 3: Substituting and Simplifying for \(g(-x)\)

Finally, we substitute \(-x\) for \(x\) in the function: \(g(-x) = (-x)^{2} - 10(-x) - 3 = x^{2} + 10x - 3\).

Key Concepts

Polynomial Function Simplification

Simplifying polynomial functions is a fundamental skill in algebra that involves reducing an expression to its simplest form. Polynomial functions, like the one given by the function \( g(x) = x^2 - 10x - 3 \), are made up of terms that include variables raised to whole number exponents and their coefficients.

Let's break down the simplification process:

• Combine like terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting the coefficients.
• Apply the distributive property: When dealing with a binomial squared, like \( (x+2)^2 \), you need to use the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand it.
• Be mindful of negative signs: When a negative sign is applied to a variable, such as in \( g(-x) \), it must be distributed across the terms within the parenthesis.

For instance, in the step-by-step solution provided, the simplification of \( g(x+2) \) involves expanding the square and then distributing the multiplication across the terms, which demonstrates these principles in action.

Substitution Method

The substitution method is utilized to evaluate functions at specific points or expressions. It's like replacing a placeholder in the function with a given value or expression to see what the output would be.

In our exercise, this involves replacing the independent variable, \( x \), with a given number or another expression. For example, when evaluating \( g(-1) \), we substitute \( -1 \) for every instance of \( x \) and then simplify the result. When we evaluate \( g(x+2) \), we substitute \( x+2 \) for \( x \), which requires extra steps like expanding and simplifying after the substitution.

This method is incredibly useful because it allows us to calculate the output for any input, which helps in understanding how the function behaves for different values of the variable.

Quadratic Function

A quadratic function is a type of polynomial function that has the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). This function generates a U-shaped curve known as a parabola. In the provided exercise, the function \( g(x) = x^2 - 10x - 3 \) is a quadratic function with \( a = 1 \), \( b = -10 \), and \( c = -3 \).

Quadratic functions are essential in algebra as they appear frequently in various contexts, from physics to economics. They can model projectile motion, economic profit, and are useful for finding maximum or minimum values. Simplifying quadratic functions, as shown in the step-by-step solution, is critical for analyzing their properties and understanding their behavior graphically and numerically. When simplified, the quadratic function's vertex, axis of symmetry, and intercepts can be grasped more readily, underscoring the importance of mastering simplification techniques.