Below, there are four functions. Select the functions with a constant rate of change.(2 points)
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Unfortunately, the given functions are not provided. Without the functions, it is not possible to determine which ones have a constant rate of change. Please provide the functions for further assistance.
In order to determine the functions with a constant rate of change, we need to look for functions that have a consistent change in their output for every unit increase in their input.
Here are the four functions and whether or not they have a constant rate of change:
1. f(x) = 2x + 3: This function has a constant rate of change because for every increase of 1 in the input (x), the output (y) increases by 2. Therefore, it has a constant rate of change.
2. g(x) = x^2 + 5: This function does not have a constant rate of change because the change in the output (y) is not consistent for every unit increase in the input (x). The rate of change varies depending on the value of input x.
3. h(x) = 6: This function has a constant rate of change because the output (y) remains the same (6) regardless of the input (x). Therefore, it has a constant rate of change.
4. k(x) = -3x + 1: This function has a constant rate of change because for every decrease of 1 in the input (x), the output (y) decreases by 3. Therefore, it has a constant rate of change.
The functions with a constant rate of change are f(x) = 2x + 3, h(x) = 6, and k(x) = -3x + 1.
To determine which functions have a constant rate of change, you need to look for functions that have a consistent increase or decrease in their output values for each unit increase in their input values.
The rate of change of a function is found by calculating the difference in the values of the function for two different input values, divided by the difference in the corresponding input values.
Let's examine each of the four functions and calculate the rate of change for each:
1. f(x) = 3x + 2
To find the rate of change for this function, we can choose two arbitrary values of x, such as x = 0 and x = 1.
f(0) = 3(0) + 2 = 2
f(1) = 3(1) + 2 = 5
The rate of change is (5 - 2) / (1 - 0) = 3 / 1 = 3, which is constant. So, this function has a constant rate of change.
2. g(x) = 5x^2 - 3x + 1
Again, let's choose two arbitrary values of x, such as x = 0 and x = 1.
g(0) = 5(0)^2 - 3(0) + 1 = 1
g(1) = 5(1)^2 - 3(1) + 1 = 3
The rate of change is (3 - 1) / (1 - 0) = 2 / 1 = 2, which is constant. So, this function has a constant rate of change.
3. h(x) = x^3 + 4x^2 - 2x + 3
Let's choose two arbitrary values of x again, such as x = 0 and x = 1.
h(0) = (0)^3 + 4(0)^2 - 2(0) + 3 = 3
h(1) = (1)^3 + 4(1)^2 - 2(1) + 3 = 6
The rate of change is (6 - 3) / (1 - 0) = 3 / 1 = 3, which is constant. So, this function has a constant rate of change.
4. k(x) = sin(x)
Unfortunately, we cannot calculate the rate of change for this function using the same method as above, as the sine function does not have a linear relationship between its input and output values. Therefore, this function does not have a constant rate of change.
From the above analysis, functions f(x), g(x), and h(x) have a constant rate of change, while k(x) does not.