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Chapter 5: Problem 7

For each polynomial function, find (a) \(f(-1)\) and \((b) f(2)\). \(f(x)=-x^{2}+2 x^{3}-8\)

Short Answer

Expert verified
f(-1) = -11, f(2) = 4

Step by step solution

01

- Identify the Given Polynomial Function

The given polynomial function is: \[f(x) = -x^2 + 2x^3 - 8\]
02

- Substitute \boldsymbol{x = -1} into the Polynomial Function

To find \( f(-1) \), substitute \( x = -1 \) into the polynomial: \[f(-1) = -(-1)^2 + 2(-1)^3 - 8\]
03

- Simplify the Equation for \boldsymbol{f(-1)}

Simplify the terms: \(-(-1)^2 = -1\), \(2(-1)^3 = -2\)So, \[f(-1) = -1 + (-2) - 8 = -1 - 2 - 8 = -11\]
04

- Substitute \boldsymbol{x = 2} into the Polynomial Function

To find \( f(2) \), substitute \( x = 2 \) into the polynomial: \[f(2) = -(2)^2 + 2(2)^3 - 8\]
05

- Simplify the Equation for \boldsymbol{f(2)}

Simplify the terms: \(-(2)^2 = -4\), \(2(2)^3 = 16\)So, \[f(2) = -4 + 16 - 8 = 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial function
A polynomial function is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. An example would be the polynomial function given in this exercise:
  • It's a combination of three terms: -x^2, 2x^3, and -8.
  • The variable is x, and the exponents (powers) are integers.
  • The coefficients (numbers multiplying the variables) are -1, 2, and -8 respectively.
function evaluation
Function evaluation involves finding the value of a function for a specific value of its variable. This is what we are doing when we find f(-1) and f(2) in this example. Evaluating a function means substituting every occurrence of the variable with the given number and performing the arithmetic operations required.This is a crucial step in understanding how the function behaves for different values of x.
substitution method
The substitution method is used during function evaluation where you replace the variable (x) with a given number.
  • In this exercise, we substitute x = -1 first: \[ f(-1) = -(-1)^2 + 2(-1)^3 - 8 \]

    Note that -(-1)^2 means squaring -1 first (which gives 1), and then taking the negative of that result (making it -1).
  • Each time -1 replaces x in the function.

  • Next, we do a similar substitution for x = 2:\[ f(2) = -(2)^2 + 2(2)^3 - 8 \]

Consistency in applying this method helps ensure we get the right result.
simplification
Simplification is the process of making an expression easier to work with. After substitution, each term needs to be simplified to reach the final value. Let's see how it is done for each substitution:
  • For f(-1):
    1. Evaluate powers first: \[ -(-1)^2 = -1 \] \[ 2(-1)^3 = -2 \]
    2. Perform operations: \[ -1 + (-2) - 8 = -11 \]
  • For f(2):
    1. Evaluate powers first: \[ -(2)^2 = -4 \] \[ 2(2)^3 = 16 \]
    2. Perform operations: \[ -4 + 16 - 8 = 4 \]
Simplification makes it easier to get to the final answer, reducing the complexity of the function step by step.

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