Problem 29 - 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. $$g(3)$$

Step-by-Step Solution

Answer

The short answer is \(g(3) = -12\).

Step 1: Substitute the value of x into the function g(x)

To find the value of \(g(3)\), we substitute \(x = 3\) in the function \(g(x) = x^2 - 4x - 9\): \[g(3) = (3)^2 - 4(3) - 9\]

Step 2: Simplify the expression

Now we just need to compute the expression we got for \(g(3)\): \[g(3) = 9 - 12 - 9\] Combine constants: \[g(3) = -3 - 9\] \[g(3) = -12\] So, the value of \(g(3)\) is -12.

Key Concepts

Function Evaluation

In mathematics, evaluating a function means finding the output value of a function for a specific input value. The process is simple: plug the input (often represented as "x") into the function and calculate the result. This tells us what the function "does" to that specific input.
For example, given the function \( g(x) = x^2 - 4x - 9 \), evaluating \( g(3) \) involves substituting \( x \) with \( 3 \).
This substitution then requires completing the expression:

• Replace \( x \) with \( 3 \) in each occurrence.
• Calculate the result: \( (3)^2 - 4(3) - 9 \).
• Final calculation shows \( 9 - 12 - 9 = -12 \).

The function evaluation has allowed us to see the specific outcome of \( g(x) \) when \( x = 3 \), resulting in \( -12 \).

Polynomial Functions

Polynomial functions are expressions that include variables raised to positive integer powers and their coefficients. These are among the most common types of functions encountered in algebra.
A polynomial in its general form might be expressed as: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]where each \( a_i \) represents a coefficient, and the highest power, \( n \), determines the degree of the polynomial.
The function \( g(x) = x^2 - 4x - 9 \) is a polynomial function of degree 2 (quadratic), because the highest exponent is 2.
Some key characteristics of polynomial functions include:

• Smooth and continuous graphs.
• Degrees that influence the number of possible roots.
• Symmetrical properties in certain cases like even or odd degree functions.

Polynomial functions are foundational in algebra and are vital for understanding more complex mathematical concepts.

Simplifying Expressions

Simplifying mathematical expressions is an essential skill that involves reducing expressions to their simplest form. This often means combining like terms and performing arithmetic operations.
Let's take an example from the solution where we simplified \( g(3) = 9 - 12 - 9 \).
We can break down the process:

• Compute the powers and products first: here, \( (3)^2 = 9 \) and \( -4(3) = -12 \).
• Combine all like terms: sum up constants in equations.
• For \( 9 - 12 - 9 \), calculate step by step: \( 9 - 12 = -3 \), then \(-3 - 9 = -12 \).

Through simplifying, we find the expression's most concise form, rendering complex expressions easier to understand and work with.