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Chapter 5: Problem 2

For each polynomial function, find (a) \(f(-1)\) and \((b) f(2)\). \(f(x)=-2 x+5\)

Short Answer

Expert verified
f(-1) = 7 and f(2) = 1.

Step by step solution

01

Title - Substitute -1 into the function

To find \(f(-1)\), substitute \-1\ for \(x\) in the polynomial \(f(x) = -2x + 5\). This gives \ f(-1) = -2(-1) + 5 \.
02

Title - Simplify the equation for f(-1)

Calculate the above expression: \ -2(-1) + 5 = 2 + 5 = 7 \. Therefore, \(f(-1) = 7\).
03

Title - Substitute 2 into the function

To find \(f(2)\), substitute \2\ for \(x\) in the polynomial \(f(x) = -2x + 5\). This gives \ f(2) = -2(2) + 5 \.
04

Title - Simplify the equation for f(2)

Calculate the above expression: \ -2(2) + 5 = -4 + 5 = 1 \. Therefore, \(f(2) = 1\).

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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 type of function that involves only non-negative integer powers of the variable. The function expressed in the exercise, \(f(x)=-2x+5\), is a linear polynomial function. Linear polynomials are the simplest form of polynomials and are characterized by their degree, which is the highest power of the variable in the function. In this case, the degree of the polynomial is 1, as the variable \(x\) is raised to the first power.
Polynomial functions can be classified into:
  • Linear Polynomials (degree 1)
  • Quadratic Polynomials (degree 2)
  • Cubic Polynomials (degree 3)
Understanding the type and degree of the polynomial helps us to evaluate it effectively at different points using appropriate methods.
Substitution Method
The substitution method involves replacing the variable in the polynomial function with a given value. For instance, the problem requires calculating \(f(-1)\) and \(f(2)\) for the polynomial function \(f(x)=-2x+5\). Here's how we do the substitution:
1. To find \(f(-1)\), replace \(x\) with \(-1\) in the equation:\[ f(-1) = -2(-1) + 5.\]2. For \(f(2)\), replace \(x\) with \(2\):\[ f(2) = -2(2) + 5.\]Substitution is a crucial step in evaluating polynomial functions because it allows us to find specific values of the function by processing the arithmetic involving constants and coefficients properly.
Algebraic Simplification
Once the variable is substituted in the polynomial function, the next step is algebraic simplification. This means performing the mathematical operations in the equation to simplify it to a single value. Let's see how it's done with our examples:
1. For \(f(-1)\):
Replace \(-1\) and simplify:\[-2(-1) + 5 = 2 + 5 = 7.\]So, \(f(-1)=7\).
2. For \(f(2)\):
Replace \(2\) and simplify:\[-2(2) + 5 = -4 + 5 = 1.\]Therefore, \(f(2)=1\).
These steps demonstrate how substitution followed by algebraic simplification assists in finding the values of a polynomial function at different points. Mastering these skills allows for accurate and efficient problem-solving in algebra.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(-3 y^{3} x^{3}\right)\left(-4 y^{4} x^{2}\right)\left(x^{2}\right)^{-4}}{18 x^{3} y^{2}\left(y^{3}\right)^{3}\left(x^{3}\right)^{-2}} $$

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