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 and the output or 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 :
We are given the following data for :
0 | 3 | 6 | 10 | 15 | |
---|---|---|---|---|---|
0 | 3 | 35 | 57 | 84 |
Now, let's find the first differences (i.e., the differences between consecutive values):
As the first differences are not constant, the function is not linear.
2. For :
We are given the following data for :
0 | 3 | 6 | 10 | 15 | |
---|---|---|---|---|---|
-1 | 20 | 41 | 69 | 104 |
Let's now find the first differences for :
Again, the first differences are not constant, meaning that is also not linear.
Conclusion:
Neither function nor is linear.
Would you like further assistance on understanding this, or help with another question?
Here are 5 related questions you could explore:
- How do we determine if a function is quadratic or cubic based on data?
- What methods can be used to find the equation of a quadratic function?
- How do you identify patterns in data to recognize non-linear relationships?
- What is the significance of first and second differences in identifying polynomial functions?
- 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
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