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Chapter 3: Problem 121

State whether the function is linear or quadratic. $$f(x)=4-5 x$$

Short Answer

Expert verified
The function is linear.

Step by step solution

01

Identify the form of the function

Examine the function given. The function is presented as: \[ f(x) = 4 - 5x \]Identify the highest power of the variable \(x\).
02

Determine the highest power of the variable

A linear function has the highest power of \(x\) as 1 (i.e., \(x^1\)). A quadratic function has the highest power of \(x\) as 2 (i.e., \(x^2\)). In the given function, the highest power of \(x\) is \(1\).
03

Conclusion

Since the highest power of \(x\) in the function \( f(x) = 4 - 5x \) is 1, the function is linear.

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Key Concepts

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

Function Identification
Identifying the type of a function is crucial in understanding its behavior and properties. Functions can be linear, quadratic, cubic, and so on, based on the highest power of their variable.
In the exercise, our focus is to determine whether the given function is linear or quadratic. The given function is: \[ f(x) = 4 - 5x \]
To identify the function type, we need to look closely at the variable's power. Here, the highest power of the variable (\
Highest Power of Variable
The highest power of the variable in a function helps to classify the function type.
For a function to be linear, the highest power of the variable (x) must be 1. In mathematical terms, this means the variable is represented as \(x^1\). For a function to be quadratic, the highest power of the variable must be 2, represented as \(x^2\).
Analyzing our function \[ f(x) = 4 - 5x \] the highest power of the variable \(x\) is 1, since it is written as \(-5x\) which implies \(-5x^1\). This information alone tells us that the function is linear.
Linear vs Quadratic Functions
Understanding the difference between linear and quadratic functions will help you solve various problems easily.
  • **Linear functions** have the form \[ f(x) = ax + b \] They graph as straight lines and are defined by a constant rate of change (slope). Linear functions have a variable with a maximum power of 1.
  • **Quadratic functions** have the form \[ f(x) = ax^2 + bx + c \] They graph as parabolas, which can open upwards or downwards. Quadratic functions have a variable with a maximum power of 2.

In the exercise, the function \[ f(x) = 4 - 5x \] fits the form of a linear function because the highest power of \(x\) is 1. Hence, we conclude that the function is linear.

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Most popular questions from this chapter

A stone is thrown directly upward from a height of 30 ft with an initial velocity of \(60 \mathrm{ft} / \mathrm{sec} .\) The height of the stone, in feet, \(t\) seconds after it has been thrown is given by the function \(s(t)=-16 t^{2}+60 t+30 .\) Determine the time at which the stone reaches its maximum height and find the maximum height.

A model rocket is launched with an initial velocity of \(120 \mathrm{ft} / \mathrm{sec}\) from a height of \(80 \mathrm{ft}\). The height of the rocket, in feet, \(t\) seconds after it has been launched is given by the function \(s(t)=-16 t^{2}+120 t+80 .\) Determine the time at which the rocket reaches its maximum height and find the maximum height.

Determine whether the function is even, odd, or neither even nor odd. $$f(x)=4 x^{2}+2 x-3$$

Determine whether the statement is true or false. The conjugate of a sum is the sum of the conjugates of the individual complex numbers.

Solve. $$x^{2 / 3}=x$$
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