Problem 14 - Algebra for College Students

Question

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ g(2) $$

Step-by-Step Solution

Answer

1

Step 1: Substitute 2 into g(x) Definition

To find the value of the function \( g(x) \) at \( x = 2 \), substitute 2 into the function's formula: \( g(x) = x^3 - 3x^2 + 5 \). This gives: \[ g(2) = 2^3 - 3(2^2) + 5 \]

Step 2: Simplify the Expression

Calculate the expression step by step: \( 2^3 = 8 \), \( 3(2^2) = 3(4) = 12 \). Substitute these values back into the equation: \[ g(2) = 8 - 12 + 5 \]

Step 3: Final Calculation

Combine the calculated terms to find the final value: \[ g(2) = 8 - 12 + 5 = 1 \]

Key Concepts

polynomial functions

A polynomial function is a type of function that involves only non-negative integer powers of the variable. Specifically, it has the form: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where the coefficients \( a_n, a_{n-1}, ..., a_0 \) are constants. For example, in the function given: \( g(x) = x^3 - 3x^2 + 5 \), the term \( x^3 \) has a coefficient of 1, \( -3x^2 \) has a coefficient of -3, and the constant term is 5. Polynomial functions are widely used because they are simple yet powerful, capable of modeling a wide range of behaviors. They are continuous and differentiable, making them easy to work with in calculus.

substitution method

The substitution method is a straightforward technique used to evaluate functions. By replacing the variable in the function with a given number, you can find the function's value at that point. Let's look at the example given: You want to find \( g(2) \). To do this, substitute 2 into the function \( g(x) = x^3 - 3x^2 + 5 \). This results in a new expression: \[ g(2) = 2^3 - 3(2^2) + 5 \]. Substitution means you replace every instance of 'x' with '2'. It's a fundamental method often used in algebra and calculus to make functions more manageable.

simplification steps

After substitution, simplification is the next crucial step. It involves calculating and reducing the expression to its simplest form. For the given function \( g(x) = x^3 - 3x^2 + 5 \), consider the expression after substitution: \( g(2) = 2^3 - 3(2^2) + 5 \). First, compute the powers and products: \( 2^3 = 8 \) \( 3(2^2) = 3(4) = 12 \) Now, substitute these back in: \[ g(2) = 8 - 12 + 5 \]. Simplification helps break down complex expressions, making them easier to handle and interpret.

final calculation

The final calculation involves combining the simplified terms to get the result. Let's follow through with our example: After simplification, we have: \( g(2) = 8 - 12 + 5 \). Perform the arithmetic operations step by step: First, subtract 12 from 8: \( 8 - 12 = -4 \). Then, add 5 to the result: \[ -4 + 5 = 1 \]. So, the final value of \( g(2) \) is 1. Final calculations are essential as they provide the ultimate result of the function evaluation.