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Chapter 9: Problem 37

If \(f(x)=5 x-2\), find \(f(0), f(2), f(-1)\), and \(f(-4)\).

Short Answer

Expert verified
\( f(0) = -2 \), \( f(2) = 8 \), \( f(-1) = -7 \), \( f(-4) = -22 \).

Step by step solution

01

Understanding the Function

The problem provides the function \( f(x) = 5x - 2 \). The goal is to find the values of this function at specific points: \( f(0) \), \( f(2) \), \( f(-1) \), and \( f(-4) \). This means we need to substitute these values into the function to determine the outputs.
02

Calculate \( f(0) \)

To find \( f(0) \), substitute \( x = 0 \) into the function: \[f(0) = 5(0) - 2 = 0 - 2 = -2\] So, \( f(0) = -2 \).
03

Calculate \( f(2) \)

To determine \( f(2) \), substitute \( x = 2 \) in the function: \[f(2) = 5(2) - 2 = 10 - 2 = 8\] Thus, \( f(2) = 8 \).
04

Calculate \( f(-1) \)

For \( f(-1) \), use \( x = -1 \): \[f(-1) = 5(-1) - 2 = -5 - 2 = -7\] Hence, \( f(-1) = -7 \).
05

Calculate \( f(-4) \)

Now let's find \( f(-4) \) by substituting \( x = -4 \) into the function: \[f(-4) = 5(-4) - 2 = -20 - 2 = -22\] Therefore, \( f(-4) = -22 \).

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Key Concepts

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

Function Evaluation
In mathematics, function evaluation is the process of determining the output of a given function at specific input values. For a linear function like \( f(x) = 5x - 2 \), the goal is to plug in different values of \( x \) to see how it affects the output. Function evaluation is a crucial aspect of understanding how functions behave and is often the first step in many mathematical problems involving functions.

Here鈥檚 a simple formula:

  • Identify the function, for example, \( f(x) = 5x - 2 \).
  • Substitute the input value in place of \( x \) in the function.
  • Perform the arithmetic operation to get the corresponding output, which is \( f(x) \).
By evaluating different inputs such as \( f(0), f(2), f(-1), \) and \( f(-4) \), you paint a clearer picture of how the function behaves over various intervals.
Substitution Method
The substitution method is a technique used to solve and evaluate functions, especially in algebraic equations. It involves replacing a variable with a specific value and then simplifying the expression. This method allows you to calculate the output for each given value of \( x \) in a function.

When approaching a problem:
  • Take the original function; for example, \( f(x) = 5x - 2 \).
  • Decide on the \( x \) value you wish to evaluate, such as \( x = 0 \).
  • Substitute \( x \) with this value in the function equation.
  • Replace every instance of \( x \) in the function with the value chosen and solve.
This operation simplifies the function to produce a single numeric output, demonstrating how altering the input impacts the outcome.
Algebraic Expressions
Algebraic expressions like \( 5x - 2 \) involve variables and coefficients and are the backbone of algebra. Understanding algebraic expressions is essential as it allows us to create functions, which in turn help us model real-world situations in mathematical terms.

An algebraic expression includes:
  • Variables, which are placeholders represented by letters like \( x \).
  • Coefficients, which are numbers multiplying the variables, such as 5 in \( 5x \).
  • Constants, which are fixed numbers, like -2 in our example.
By changing the value of the variable, you can see different outputs, making algebraic expressions a versatile tool for problem-solving. Performing operations, such as function evaluations, involves manipulating these expressions according to algebraic rules and principles.

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Most popular questions from this chapter

If \(y\) is inversely proportional to \(x\), and \(y=\frac{1}{9}\) when \(x=12\), find the value of \(y\) when \(x=8\).

Suppose that \(y\) varies inversely as \(x\). Does doubling the value of \(x\) also double the value of \(y\) ? Explain your answer.

Determine the indicated functional values. (Objective 2 ) If \(f(x)=\sqrt{3 x-2}\) and \(g(x)=-x+4\), find \((f \circ g)(1)\) and \((g \circ f)(6)\).

Find the constant of variation for each of the stated conditions. \(y\) varies directly as the square of \(x\), and \(y=-144\) when \(x=6 .\)

Translate each statement of variation into an equation, and use \(k\) as the constant of variation. \(y\) varies inversely as the square of \(x\).
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