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Chapter 3: Problem 34

For each function \(f,\) find \((a) f(2),(b) f(0),\) and \((c) f(-3) .\) See Example 5 $$ f(x)=-3 x+5 $$

Short Answer

Expert verified
f(2) = -1, f(0) = 5, f(-3) = 14.

Step by step solution

01

- Understand the Function

The given function is \( f(x) = -3x + 5 \). This means for any input \( x \), the output \( f(x) \) can be calculated by multiplying \( x \) by \( -3 \) and then adding \( 5 \).
02

- Calculate \( f(2) \)

Substitute \( 2 \) for \( x \) in the function \( f \): \[ f(2) = -3(2) + 5 \] Simplify the expression: \[ f(2) = -6 + 5 = -1 \].
03

- Calculate \( f(0) \)

Substitute \( 0 \) for \( x \) in the function \( f \): \[ f(0) = -3(0) + 5 \] Simplify the expression: \[ f(0) = 0 + 5 = 5 \].
04

- Calculate \( f(-3) \)

Substitute \( -3 \) for \( x \) in the function \( f \): \[ f(-3) = -3(-3) + 5 \] Simplify the expression: \[ f(-3) = 9 + 5 = 14 \].

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Key Concepts

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

Understanding Linear Functions
A linear function is one where each input is multiplied by a constant and then another constant is added. For example, in the function given in the exercise, \texpressed as \[ f(x) = -3x + 5 \], the term \[ -3x \] shows that each input \( x \) is multiplied by \( -3 \),\tand then 5 is added to that product. The general formula for a linear function is \[ f(x) = mx + b \], where \( m \) represents the slope and \( b \) is the y-intercept. Linear functions are important because:\t
    \t
  • They grow at a constant rate, making them predictable and easy to graph.
    • \t
  • They can represent many real-world situations, such as calculating costs or speeds that change linearly.
    • \t
\tUnderstanding this concept helps in recognizing patterns and predicting outcomes from given inputs. They are helpful in various mathematical and practical applications.\t
\t
Substitution Method
Substitution is a technique where we replace one variable with a specific value to find the outcome. In the given function \[ f(x) = -3x + 5 \], to find \[ f(2) \], \twe substitute 2 into the function in place of \( x \). This means that \[ f(2) = -3(2) + 5 \]. Here, the variable \( x \) is replaced with 2, and then the expression is solved step by step. By replacing the variable with a given number:\t
    \t
  • We can directly find the function's value at specific points.
    • \t
  • This method makes it easier to calculate outcomes without graphing.
    • \t
\tLet's check our results from the steps provided:\t
    \t
  • For \( f(2) \): Substituted 2, the result is \[ f(2) = -6 + 5 = -1 \].
    • \t
  • For \( f(0) \): Substituted 0, the result is \[ f(0) = 0 + 5 = 5 \].
    • \t
  • For \( f(-3) \): Substituted -3, the result is \[ f(-3) = 9 + 5 = 14 \].
    • \t
\tUnderstanding substitution aids in analyzing and calculating functions without visual aids.\t
\t
Simplifying Expressions
Simplification involves reducing an expression to its simplest form. This means combining like terms, applying arithmetic operations, and making the expression as compact as possible. \tFor the function \[ f(x) = -3x + 5 \], after substituting the variable with a number, like \[ f(2) = -3(2) + 5 \], simplifying involves:\t
    \t
  • Multiplying \( -3 \times 2 \), which gives -6.
    • \t
  • Then adding 5 to -6, resulting in -1.
    • \t
\tThe steps are straightforward but require attention to detail to avoid mistakes. Following these correctly provides you the exact value of the function for any given input. Simplification helps in:\t
    \t
  • Clearly understanding the result of a function.
    • \t
  • Ensuring that the calculations are accurate and errors minimized.
    • \t
  • Simplifying and solving complex mathematical problems easily.
    • \t
\tGrasping this concept improves your ability to handle algebraic expressions promptly and effectively.\t

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Most popular questions from this chapter

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example \(6 .\) $$ \begin{aligned} &2 x+5 y=4\\\ &4 x+10 y=1 \end{aligned} $$

What is the slope of a line whose graph is parallel to the graph of \(3 x+y=7 ?\) Perpendicular to the graph of \(3 x+y=7 ?\)

Do \((3,4)\) and \((4,3)\) correspond to the same point in the plane? Explain.

Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (3,5),(3,0), \text { and }(3,-3) $$

Plot and label each point in a rectangular coordinate system. $$ (5,3) $$
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