Final answer:
The function f(x) = 1/4x + 7 has a positive slope, and for any positive value of x, f(x) will be greater than zero due to the positive slope and the addition of 7. Therefore, the correct statement is: If x > 0, then f(x) > 0.
Step-by-step explanation:
If we consider the function f(x) = \frac{1}{4}x + 7, we are looking to determine which statement about the function's value based on the value of x is always true. First, let's analyze the function: it's a linear function with a positive slope of \frac{1}{4} which means that as x increases, f(x) also increases.
Now let's check the given options one by one:
- a) f(x) > 0, then f(x) > 0: This option is somewhat redundant and doesn't provide a logical conditional statement, so we can dismiss it.
- b) If x > 0, then f(x) > 0: Given that the slope is positive, for any positive value of x, the function will indeed produce a result that is greater than zero, since 7 is added to a positive number. This looks like a strong candidate.
- c) f(x) > 0: This statement is not necessarily true for all values of x, especially when x can be negative.
- d) If f(x) < 0, then f(x) < 0: This is similar to option a) in that it repeats itself and does not provide a conditional relationship between x and f(x).
Therefore, the correct answer is b) If x > 0, then f(x) > 0.