Problem 40 - Algebra 2

Question

Find each value if \(f(x)=3 x-5\) and \(g(x)=x^{2}-x\) \(g(5 n)\)

Step-by-Step Solution

Answer

\(g(5n) = 25n^2 - 5n\)

Step 1: Understand the function g(x)

The function given is \(g(x) = x^2 - x\). This is a quadratic function and involves squaring the input \(x\) and then subtracting \(x\) from the result.

Step 2: Substitute the expression into g(x)

We need to evaluate \(g(5n)\). This means that wherever you see \(x\) in the function \(g(x)\), you will replace it with \(5n\). So, the function changes to \(g(5n) = (5n)^2 - 5n\).

Step 3: Perform the Calculations

Calculate \((5n)^2\) to get \(25n^2\). Then calculate \(-5n\). Therefore, \(g(5n) = 25n^2 - 5n\).

Key Concepts

Function Evaluation

When dealing with functions, understanding how to evaluate them is crucial. Function evaluation refers to finding the value of a function by substituting the input, often represented by a variable, with a given number or expression.
For example, consider a function, such as \( f(x) = 3x - 5 \). To evaluate \( f(x) \) at a specific point, say \( x = 2 \), you substitute \( 2 \) for \( x \) to obtain \( f(2) = 3(2) - 5 = 6 - 5 = 1 \).
This process enables you to determine the output of a function for various inputs, allowing for practical applications across various mathematical and real-world scenarios.

• Identify the function and the input.
• Substitute the input value into the function.
• Simplify the expression to find the result.

Function evaluation allows you to grasp how changes in input affect the output, helping make predictions or analyze behaviors.

Substitution Method

In mathematics, substitutions play a vital role in simplifying expressions and solving equations.
Specifically, in function evaluation, the substitution method involves replacing a variable with another expression or value to determine the outcome of the function.
For example, with the function \( g(x) = x^2 - x \), if you are asked to evaluate \( g(5n) \), it involves a substitution where \( x \) is replaced with \( 5n \).
Here’s how you do it:

• Identify where the substitution is to be made, i.e., the variable in the function.
• Replace this variable with the given expression (or value) wherever it appears.
• Simplify the resulting expression to get your final answer.

Using substitutions can transform complex equations into forms that are easier to work with, facilitating straightforward computations.

Polynomials

Polynomials are expressions made from variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
A standard form of a polynomial looks like this: \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where each \( a_i \) represents a coefficient.
In the exercise, the function \( g(x) = x^2 - x \) is a quadratic polynomial because the highest power of \( x \) is 2. Quadratic polynomials take the form \( ax^2 + bx + c \).

• Degree: The highest exponent, which in this case is 2, making it a quadratic.
• Coefficients: The numbers multiplying the variables, which are 1 and -1 for \( x^2 \) and \( x \), respectively.
• Terms: Each part of the polynomial separated by plus or minus signs, which here are \( x^2 \) and \( -x \).

Quadratics and other polynomials appear commonly in algebra and calculus, making understanding them valuable for further mathematical studies.