Final answer:
The exponential and linear functions y = 2x² and y = 2x are compared. The statements about the comparison are evaluated and explained.
Step-by-step explanation:
The given functions are y = 2x² and y = 2x. To compare the functions, we can graph them and analyze their behaviors.
First, let's compare the y-values for different x-values. For the exponential function y = 2x², the y-values will always be greater than or equal to zero, as squaring a number always results in a non-negative value. On the other hand, for the linear function y = 2x, the y-values can be positive, negative, or zero. Therefore, the statement 'For any x-value, the y-value of the exponential function is always greater' is false.
Next, let's consider the statement 'For any x-value, the y-value of the exponential function is always smaller.' This statement is also false, as the exponential function y = 2x² can produce larger y-values than the linear function y = 2x for certain x-values, especially as x grows larger.
From the above analysis, we can conclude that the statements 'For some x-values, the y-value of the exponential function is smaller' and 'For some x-values, the y-value of the exponential function is greater' are both true.
Regarding the statement 'For any x-value greater than 7, the y-value of the exponential function is greater,' it is false because the linear function y = 2x will have larger y-values for sufficiently large x-values.
Finally, the last statement 'For equal intervals, the y-values of both functions have a common ratio' is not applicable here since the functions given are not in a ratio form.
Learn more about Comparison of exponential and linear functions