Which functions have an additive rate of change of 3?
Understand the Problem
The question is asking which mathematical functions exhibit an additive rate of change of 3, meaning these functions should increase by 3 units for every unit increase in their input.
Answer
Functions of the form $f(x) = 3x + b$, where $b$ is any constant, exhibit an additive rate of change of 3.
Answer for screen readers
The functions exhibiting an additive rate of change of 3 can be expressed as:
$$ f(x) = 3x + b $$
where $b$ is any constant.
Steps to Solve
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Define Additive Rate of Change
The additive rate of change of a function indicates how much the output of the function changes as the input changes by a certain amount. Here, we are looking for functions where for every increase of 1 in the input, the output increases by 3. -
Identify the Function Characteristics
A function exhibiting an additive rate of change of 3 can be expressed in the form:
$$ f(x) = 3x + b $$
where $b$ is any constant. This indicates that the slope (rate of change) of the function is 3. -
Verify with Examples
Consider a specific example:
For the function $f(x) = 3x + 2$, we can test the additive rate of change.
- If $x$ increases from $a$ to $a + 1$, then:
$$ f(a + 1) = 3(a + 1) + 2 = 3a + 3 + 2 = 3a + 5 $$
$$ f(a) = 3a + 2 $$
The change in function value would be:
$$ f(a + 1) - f(a) = (3a + 5) - (3a + 2) = 3 $$
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General Form of Functions with Additive Rate of Change 3
From our discussion, it's clear that all functions of the form $f(x) = 3x + b$ (where $b$ is any real number) will exhibit an additive rate of change of 3.
The functions exhibiting an additive rate of change of 3 can be expressed as:
$$ f(x) = 3x + b $$
where $b$ is any constant.
More Information
Functions with a linear relationship exhibit constant rates of change. The additive rate of change means that the function consistently increases by a specified amount for each unit increase in the input, making these functions predictable and easy to work with in various applications.
Tips
- Assuming all functions have constant rates of change: Only linear functions exhibit constant additive rates of change. Non-linear functions may have varying rates.
- Forgetting the constant term: The presence of $b$ indicates that the function can still have different starting points but maintains the same rate.
