Math Problem Statement

Decide which of the two given functions is linear and find its equation. x 0 3 6 10 15 f(x) 0 3 35 57 84 g(x) −1 20 41 69 104

Solution

To determine which of the two functions is linear, we need to check whether the relationship between the input xx and the output f(x)f(x) or g(x)g(x) follows a constant rate of change. This can be done by checking if the differences between consecutive values of the outputs (i.e., the first differences) are constant.

1. For f(x)f(x):

We are given the following data for f(x)f(x):

xx0361015
f(x)f(x)03355784

Now, let's find the first differences (i.e., the differences between consecutive f(x)f(x) values):

First Differences for f(x)={30,353,5735,8457}\text{First Differences for } f(x) = \{ 3-0, 35-3, 57-35, 84-57 \} {3,32,22,27}\{ 3, 32, 22, 27 \}

As the first differences are not constant, the function f(x)f(x) is not linear.

2. For g(x)g(x):

We are given the following data for g(x)g(x):

xx0361015
g(x)g(x)-1204169104

Let's now find the first differences for g(x)g(x):

First Differences for g(x)={20(1),4120,6941,10469}\text{First Differences for } g(x) = \{ 20-(-1), 41-20, 69-41, 104-69 \} {21,21,28,35}\{ 21, 21, 28, 35 \}

Again, the first differences are not constant, meaning that g(x)g(x) is also not linear.


Conclusion:

Neither function f(x)f(x) nor g(x)g(x) is linear.

Would you like further assistance on understanding this, or help with another question?


Here are 5 related questions you could explore:

  1. How do we determine if a function is quadratic or cubic based on data?
  2. What methods can be used to find the equation of a quadratic function?
  3. How do you identify patterns in data to recognize non-linear relationships?
  4. What is the significance of first and second differences in identifying polynomial functions?
  5. How can you create a linear regression model from data?

Tip: To determine if a function is linear, always check if the first differences (changes between consecutive outputs) are constant. If they are, the function is linear.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Difference of Functions
First Differences

Formulas

First Differences Formula: Δf(x) = f(x_n) - f(x_{n-1})

Theorems

Linear Function Theorem
Constant Rate of Change

Suitable Grade Level

Grades 7-9