Problem 62 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Evaluate the function for the given value of \(x\). $$g(x)=-2 x^4+7 x^3+x-2 ; x=3$$

Step-by-Step Solution

Answer

The value of function \( g(x) \) for \( x=3 \) is 28.

Step 1: Function Interpretation

First interpret the provided function, which is \( g(x)=-2 x^4+7 x^3+x-2 \). The goal is to substitute \( x=3 \) into this function and compute the resulting value.

Step 2: Substitution of variable \(x\)

Substitute \( x = 3 \) into the function: \( g(3)=-2 (3)^4+7 (3)^3+(3)-2 \).

Step 3: Simplify the function

Now simplify the function to obtain the final result: \( g(3)=-2 (81)+7 (27)+3-2 = -162+189+3-2 = 28 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, particularly when dealing with polynomial functions. It involves taking a given function and determining its output based on a specific input value.
For the provided exercise, the function is defined as:

• \( g(x) = -2x^4 + 7x^3 + x - 2 \)
• We need to find the value of this function at \( x = 3 \).

In essence, evaluating the function for \( x = 3 \) means calculating the exact result this expression yields when \( x \) takes the value 3. This process ensures that we understand how a function behaves for specific inputs, revealing trends or specific solutions of functional equations.

Substitution Method

The substitution method is a straightforward technique often used in algebra to evaluate functions. When you substitute a value into a function, you replace the variable with the given number, allowing you to perform simplification and calculation.
In the context of our exercise, this involves substituting \( x = 3 \) into the polynomial function \( g(x) = -2x^4 + 7x^3 + x - 2 \).
With substitution, it transforms into:

• \( g(3) = -2(3)^4 + 7(3)^3 + (3) - 2 \)

This direct replacement lets us convert the general expression into a specific numeric form. Substitution is pivotal when working with polynomials, as it lays the groundwork for further simplification, ultimately leading to the function's evaluation.

Polynomial Simplification

Polynomial simplification follows the substitution step and involves methodically simplifying the expression to reach the final result.
Here, the substituted polynomial is:

• \( -2(3)^4 + 7(3)^3 + (3) - 2 \)

First, compute the powers:

• \( 3^4 = 81 \) and \( 3^3 = 27 \)

Now substitute these back into the expression:

• \( -2(81) + 7(27) + 3 - 2 \)

The operations lead to:

• \( -162 + 189 + 3 - 2 \)

Breaking this further down gives:

• \( -162 + 189 = 27 \)
• Then, \( 27 + 3 = 30 \)
• Finally, \( 30 - 2 = 28 \)

So, by simplifying these steps, we find that \( g(3) = 28 \). This simplification process highlights the importance of order in arithmetic operations, revealing the function's result after substitution.