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Chapter 2: Problem 13

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(100) $$

Short Answer

Expert verified
-296

Step by step solution

01

Identify the function and the input

Determine which function needs to be evaluated and at which input. In this case, the function is \(f(x) = -3x + 4\) and the input is \(x = 100\).
02

Substitute the input value

Replace the variable \(x\) in the function \(f(x) = -3x + 4\) with the given input value \(100\).
03

Evaluate the expression

Calculate the value of the function by simplifying the expression: \[ f(100) = -3(100) + 4 \]
04

Perform multiplication

First, multiply \(-3\) and \(100\) together: \[ -3 \times 100 = -300 \]
05

Complete the final calculation

Add \(-300\) and \(4\): \[ -300 + 4 = -296 \]

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Key Concepts

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

algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to represent real-world problems in a mathematical form and solve them efficiently. In this exercise, we focus on linear functions and substitution which are key concepts in algebra.

To start, remember that functions like \(f(x) = -3x + 4\) translate a given input (x) into an output using a specific rule. In this case, the rule combines multiplication and addition. These rules are fundamental in algebra for solving equations and understanding relationships between variables.
linear functions
Linear functions are a type of algebraic expression where each term is either a constant or the product of a constant and a single variable. They form straight lines when graphed and have the general form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

In the exercise, the function \(f(x) = -3x + 4\) is a linear function. Here, \(m = -3\) and \(b = 4\). The slope \(-3\) tells us that for every unit increase in \(x\), the output decreases by 3 units. The y-intercept 4 indicates that when \(x = 0\), the output is 4.

Understanding linear functions helps in predicting the behavior of a system and solving for unknown variables.
substitution
Substitution is a key method in algebra for solving and evaluating functions. It involves replacing a variable with a specific value to work out the solution. This technique is powerful for simplifying functions and solving equations.

In the given exercise, we used substitution to find \(f(100)\). By replacing \(x\) with 100 in the function \(f(x) = -3x + 4\), we followed these steps:

  • Replace \(x\) with 100: \(f(100) = -3(100) + 4\)
  • Calculate the multiplication: \(-3 \times 100 = -300\)
  • Add the constant to the product: \(-300 + 4 = -296\)

Thus, we used substitution to find that \(f(100) = -296\). This method is essential for handling more complex functions and algebraic problems.

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Most popular questions from this chapter

Graph each linear or constant function. Give the domain and range. $$ g(x)=-4 $$

The upper deck at Guaranteed Rate Field in Chicago has produced, among other complaints, displeasure with its steepness. It is \(160 \mathrm{ft}\) from home plate to the front of the upper deck and \(250 \mathrm{ft}\) from home plate to the back. The top of the upper deck is \(63 \mathrm{ft}\) above the bottom. What is its slope?

Use your knowledge of the slopes of parallel and perpendicular lines. Show that \((-13,-9),(-11,-1),(2,-2),\) and (4,6) are the vertices of a parallelogram. (Hint: A parallelogram is a four-sided figure with opposite sides parallel.)

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

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 2 & 4 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline-1 & 1 \\ \hline-2 & 4 \end{array} $$
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