Problem 38 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$g(a+3)$$

Step-by-Step Solution

Answer

The simplified expression for \(g(a+3)\) is: $$g(a+3) = a^2 + 2a - 12$$

Step 1: Substitute a+3 into g(x)

Replace the variable x in the function \(g(x) = x^2 - 4x - 9\) with a+3: $$g(a+3) = (a+3)^2 - 4(a+3) - 9$$

Step 2: Expand (a+3)^2

To expand \((a+3)^2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\): $$(a+3)^2 = a^2 + 2(a)(3) + (3)^2 = a^2 + 6a + 9$$

Step 3: Substitute the expansion back into g(a+3)

Replace \((a+3)^2\) in the expression \(g(a+3)\) with the expanded form: $$g(a+3) = (a^2 + 6a + 9) - 4(a+3) - 9$$

Step 4: Distribute -4

Distribute the -4 to both terms inside the parenthesis: $$g(a+3) = a^2 + 6a + 9 - 4a - 12 - 9$$

Step 5: Simplify

Combine like terms in the expression: $$g(a+3) = a^2 + 6a - 4a + 9 - 12 - 9 = a^2 + 2a - 12$$ The simplified expression for \(g(a+3)\) is: $$g(a+3) = a^2 + 2a - 12$$

Key Concepts

Function Evaluation

When tackling algebraic functions, it is often necessary to evaluate the function for a certain value or expression. This means we substitute a given value or expression into the function in place of the variable.

For example, given a function such as \(g(x) = x^2 - 4x - 9\), and you want to find \(g(a+3)\), you substitute \(a+3\) wherever you see \(x\) in the function. This action is known as function evaluation.

The result is a new expression: - \((a+3)^2 - 4(a+3) - 9\).

At this stage, you have successfully replaced the variable \(x\) with \(a+3\), setting the stage for further manipulations like expansion and simplification.

Polynomial Expansion

Polynomial expansion involves spreading out expressions raised to a power, into a simpler form using algebraic identities.

In the context of the exercise, we are tasked with expanding \((a+3)^2\).

To achieve this, use the identity for the square of a binomial:

• \((a+b)^2 = a^2 + 2ab + b^2\)

Thus, for \((a+3)^2\), it expands to:

• \(a^2 + 2(a)(3) + (3)^2\)

which further simplifies to:

• \(a^2 + 6a + 9\)

This transformation from a compact notation, \((a+3)^2\), to a fully expanded polynomial form is crucial, as it allows easier combination with other parts of the expression in subsequent steps.

Simplification

Simplification involves combining like terms in an expression to create a more straightforward form. After polynomial expansion, you often end up with expressions that can be combined or further simplified.

In our example, we expanded \((a+3)^2\) to \(a^2 + 6a + 9\) and distributed \(-4\) across \(a+3\) to give \(-4a - 12\).

By plugging these into our expression, we get:

• \(a^2 + 6a + 9 - 4a - 12 - 9\)

Now, combine like terms:

• Combine \(6a\) and \(-4a\) to get \(2a\)
• Add constant terms \(9\), \(-12\), and \(-9\) to get \(-12\)

After simplification, the expression becomes:

• \(a^2 + 2a - 12\)

Simplification helps to reduce complexity by bringing similar terms together, making the expression easier to interpret or further use.

Substitution in Functions

Substitution in functions, especially in algebra, is a fundamental technique where one replaces the variable in the function with a specific number or another expression.

The purpose of substitution is often to evaluate a function or to develop a specific expression based on given parameters.

From our task, replacing \(x\) in \(g(x)\) with \(a+3\) demonstrates substitution:

• This transforms the original function \(g(x) = x^2 - 4x - 9\) to \(g(a+3) = (a+3)^2 - 4(a+3) - 9\).

Substitutions do not change the nature of the function, they simply adapt it to fit particular scenarios or allow calculations for specific cases.

Always ensure substitutions are correctly placed to prevent unintended errors in the algebraic process.