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Chapter 4: Problem 9

For each polynomial function, find (a) \(f(-1),(b) f(2),\) and \((c) f(0)\). \(f(x)=6 x-4\)

Short Answer

Expert verified
f(-1) = -10, f(2) = 8, f(0) = -4

Step by step solution

01

Evaluate the Function at x = -1

To find the value of the polynomial when x = -1, substitute -1 for x in the function. Given: o f(x) = 6x - 4 Therefore, f(-1) = 6(-1) - 4 f(-1) = -6 - 4 f(-1) = -10
02

Evaluate the Function at x = 2

To find the value of the polynomial when x = 2, substitute 2 for x in the function. Given: f(x) = 6x - 4 Therefore, f(2) = 6(2) - 4 f(2) = 12 - 4 f(2) = 8
03

Evaluate the Function at x = 0

To find the value of the polynomial when x = 0, substitute 0 for x in the function. Given: f(x) = 6x - 4 Therefore, f(0) = 6(0) - 4 f(0) = 0 - 4 f(0) = -4

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Key Concepts

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

polynomial functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, the polynomial function given in the exercise is \(f(x) = 6x - 4\). This is called a linear polynomial because the highest exponent of the variable x is 1. Polynomial functions are used extensively in mathematics to describe a variety of phenomena. They have different degrees depending on the highest power of the variable involved. Understanding how to work with polynomial functions is crucial for solving many types of algebraic problems.
function evaluation
Function evaluation involves finding the value of a function for a particular input. For example, to evaluate the function \(f(x) = 6x - 4\) at specific values of x, we simply replace x with the given number. In the exercise, we are asked to find \(f(-1)\), \(f(2)\), and \(f(0)\). This process helps us understand how the function behaves at different points. Knowing how to evaluate functions is essential for graphing them and understanding their properties.
substitution method
The substitution method is used to replace variables in an algebraic expression with specific values. This is exactly what we do when evaluating polynomial functions. For instance, to find \(f(-1)\) in the given function \(f(x) = 6x - 4\), we substitute \(-1\) in place of x: \(f(-1) = 6(-1) - 4\). This results in: \(f(-1) = -6 - 4 = -10\). The substitution method is straightforward but requires careful arithmetic to ensure accuracy. It's a foundational technique in algebra that allows us to solve problems efficiently.
basic algebra
Basic algebra involves elementary mathematical operations such as addition, subtraction, multiplication, and division, along with the manipulation of variables and constants. In the given exercise, we use basic algebra to simplify the polynomial expression after substitution. For example, in the step \(f(-1) = 6(-1) - 4\), we multiply 6 by -1 to get -6, then subtract 4 to get -10. Mastery of basic algebra is crucial for solving more complex mathematical problems, as it forms the foundation upon which more advanced concepts are built.

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Most popular questions from this chapter

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5}{4}\right)^{-2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{12 k^{-2}\left(k^{-3}\right)^{-4}}{6 k^{5}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(3 p^{-4}\right)^{2}\left(p^{3}\right)^{-1} $$

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{3 z}{4}\right)^{-3} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{2^{2} y^{4}\left(y^{-3}\right)^{-1}}{2^{5} y^{-2}} $$
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