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Chapter 2: Problem 8

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(-5) $$

Short Answer

Expert verified
19

Step by step solution

01

Title - Identify the function to use

First, identify which function should be used to find the value of the expression. In this case, we are given the function \( f(x) = -3x + 4 \) and need to find \( f(-5) \).
02

Title - Substitute the given value into the function

Substitute \( x = -5 \) into the function \( f(x) \). This means replacing every instance of \( x \) in \( f(x) \) with \(-5\). Thus, we get \( f(-5) = -3(-5) + 4 \).
03

Title - Simplify the expression

Now, simplify the expression obtained after substitution: \( -3(-5) + 4 \). Begin by multiplying: \( -3 \times -5 = 15 \). Then add the result to 4: \( 15 + 4 = 19 \).

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Key Concepts

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

Substitution
Substitution is an essential concept in algebra. It involves replacing a variable with a given number to evaluate an expression. In this exercise, we need to find the value of the function \( f(x) \) at \( x = -5 \). To do this, we substitute -5 into \( f(x) = -3x + 4 \). This means every instance of \( x \) in the function will be replaced by -5. Therefore, the function changes from \( f(x) = -3x + 4 \) to \( f(-5) = -3(-5) + 4 \). Substitution helps us understand how functions behave for different values of their variables.

Here are some tips for successful substitution:
  • Carefully identify which function you are working with.
  • Replace every instance of the variable with the given number.
  • Double-check your replacements to avoid mistakes.
Simplification
Simplification is the process of making an algebraic expression as simple as possible. After substituting \( x = -5 \) into the function \( f(x) = -3x + 4 \), we get the expression \( f(-5) = -3(-5) + 4 \). Simplification involves performing the operations in the correct order to reach the final answer.

Follow these steps for simplification:
  • Perform any multiplication or division first. In this case, multiply \( -3 \times -5 \), which equals 15.
  • Next, perform any addition or subtraction. Add 4 to the 15 obtained from the previous step: \( 15 + 4 \).
  • Combine the terms to get the simplified result: \( 19 \).
Simplification helps make equations and expressions easier to understand and solve.
Linear Functions
A linear function is a type of function that creates a straight line when graphed. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this exercise, the function \( f(x) = -3x + 4 \) is a linear function.

Characteristics of linear functions include:
  • The graph is a straight line.
  • The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises as it moves right, while a negative slope means it falls.
  • The y-intercept \( b \) indicates where the line crosses the y-axis.
Understanding linear functions is crucial in algebra as they form the foundation for more complex functions and equations.

In this exercise, the linear function helps us determine the value of \( f(x) \) for any given \( x \). By substituting and simplifying, we find that \( f(-5) = 19 \).

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Most popular questions from this chapter

Rental on a car is \(\$ 150,\) plus \(\$ 0.50\) per mile. Let \(x\) represent the number of miles driven and \(f(x)\) represent the total cost to rent the car. (a) Write a linear function that models this situation. (b) Find \(f(250)\). Interpret the answer in the context of this problem. (c) Find the value of \(x\) if \(f(x)=400\). Express this situation using function notation, and interpret it in the context of this problem.

Find the slope of each line, and sketch its graph. \(4 x-y=4\)

Graph each linear or constant function. Give the domain and range. $$ f(x)=5 $$

Graph the intersection of each pair of inequalities. $$ x-y \geq 2 \text { and } x \geq 3 $$

Graph each line passing through the given point and having the given slope. (-3,1)\(;\) undefined slope
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