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Chapter 4: Problem 10

For each polynomial function, find (a) \(f(-1),(b) f(2),\) and \((c) f(0)\). \(f(x)=-2 x+5\)

Short Answer

Expert verified
f(-1) = 7, f(2) = 1, f(0) = 5

Step by step solution

01

Evaluate \(f(-1)\)

To find \(f(-1)\), substitute \(-1\) in place of \(x\) in the polynomial function. \(f(x) = -2x + 5\) \(f(-1) = -2(-1) + 5\) Calculate the value:\(f(-1) = 2 + 5 = 7\)
02

Evaluate \(f(2)\)

Next, substitute \(2\) in place of \(x\) in the polynomial function. \(f(x) = -2x + 5\) \(f(2) = -2(2) + 5\) Calculate the value:\(f(2) = -4 + 5 = 1\)
03

Evaluate \(f(0)\)

Finally, substitute \(0\) in place of \(x\) in the polynomial function. \(f(x) = -2x + 5\) \(f(0) = -2(0) + 5\) Calculate the value:\(f(0) = 0 + 5 = 5\)

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Key Concepts

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

Function Evaluation
Let's dive into the concept of function evaluation. This process involves determining the output of a function for specific input values. In simpler terms, if you have a function, say, a polynomial function, you plug in a number for the variable (usually denoted as x) and compute the result.
This can help us understand the behavior of the function at different points.

Take our example polynomial function:
$$f(x) = -2x + 5$$
To evaluate this function at specific points, we simply substitute the values for x and simplify.
Polynomial Substitution
Polynomial substitution is essential when dealing with polynomial functions. It's a method where we replace the variable in the polynomial with a given number to find the value at that point.
For instance, in the function:
$$f(x) = -2x + 5$$
we want to find the values of:
$$f(-1), f(2), \text{and} f(0)$$
Here are the steps:
  • **f(-1)**: Replace x with -1:

$$ f(-1) = -2(-1) + 5 = 2 + 5 = 7 $$
  • **f(2)**: Replace x with 2:

$$ f(2) = -2(2) + 5 = -4 + 5 = 1 $$
  • **f(0)**: Replace x with 0:

$$ f(0) = -2(0) + 5 = 0 + 5 = 5 $$
By performing these substitutions, you can easily find the values of the polynomial at the given x values.
Linear Functions
Linear functions are fundamental in algebra. They form straight lines when plotted on a graph and have the general form:
$$ f(x) = mx + b $$
where **m** is the slope and **b** is the y-intercept.

Our given function:
$$ f(x) = -2x + 5 $$
is a linear function with a slope of -2 and a y-intercept of 5.

Linear functions have several key characteristics:
  • They grow at constant rates.
  • The slope (m) determines the steepness of the line.
  • The y-intercept (b) is the value of the function when x is 0.
To understand the function better, you can input various x values and observe the changes in the output. This practice helps in grasping the nature and behavior of linear functions.

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Most popular questions from this chapter

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