If f(x) is a continuous function defined for all real numbers, f(-1)=1, f(-5)=-10, and f(x)=0 for one and only one value of x, then which of the following could be that x value?
a) -6
b) -5
c) -4
d) 0
It has to be between -1 and -5. Are any of the answers in that range?
To solve this question, we need to consider the given information and apply the properties of continuous functions.
First, we are told that f(x) is a continuous function defined for all real numbers. This means that there are no "holes" or "jumps" in the graph of the function.
Next, we are given three specific values of the function: f(-1) = 1, f(-5) = -10, and f(x) = 0 for one and only one value of x.
From the first two values, we can see that f(-1) = 1 and f(-5) = -10. This means that the function has a positive value at x = -1 and a negative value at x = -5.
Since f(x) = 0 for one and only one value of x, we can conclude that the graph of the function must cross the x-axis (or have a "root") at some point.
Considering the given choices:
a) -6: The function already has a negative value at x = -5, so it cannot be the answer.
b) -5: The function has a negative value at x = -5, so it is a possibility.
c) -4: There is no given information about this value, so we cannot determine its suitability.
d) 0: There is no given information about this value, so we cannot determine its suitability.
Based on the above analysis, the correct answer could be b) -5. However, it is important to note that without additional information or a graph of the function, we cannot be completely certain about the answer.
First, let's consider the information given. We know that f(x) is a continuous function defined for all real numbers.
We are also given the values f(-1) = 1 and f(-5) = -10.
Since f(x) is continuous, it means that there are no breaks or gaps in the graph of the function. This implies that the function must cross the x-axis at some point where f(x) = 0.
Looking at the given answer choices:
a) -6
b) -5
c) -4
d) 0
We know that f(x) = 0 for one and only one value of x, so we can eliminate answer choices a), c), and d) because they have multiple values.
Therefore, the only possible x value where f(x) = 0 is option b) -5.