Final answer:
The true statements regarding the function f(x)=10/(x+5) are that the domain excludes x=-5, there is a vertical asymptote at x=-5, the y-intercept is at (0,2), and the end behavior is such that as x approaches negative or positive infinity, f(x) approaches 0. There are no x-intercepts, and the range is all real numbers except 0.
Step-by-step explanation:
The function given is f(x) = \frac{10}{x+5}. To analyze the properties of this function, we need to understand its behavior when x approaches certain values and its graph within the given restrictions. Let's check each statement given:
- The domain of f(x) is (-∞, -5) U (-5, ∞). This means x can be any real number except -5, where the function would be undefined due to division by zero, indicating there is a vertical asymptote at x = -5.
- The range of f(x) is all real numbers since as x changes, the value of f(x) can take any real number except 0; there are no other restrictions imposed on f(x) by the function itself.
- The x-intercept does not occur at (-5,0) because there the function is undefined. Actually, the function cannot have an x-intercept because f(x) is never zero for any real value of x.
- The y-intercept is (0,2), which can be found by plugging x=0 into the function, yielding f(0) = \frac{10}{0+5} = 2.
- There is indeed a vertical asymptote at x = -5 since the function is undefined at this point and the values of f(x) grow without bound as x approaches -5.
- The end behavior suggests that as x approaches negative or positive infinity, f(x) approaches 0. This is because the numerator remains constant, and as the denominator grows larger in magnitude, the overall value of the fraction diminishes towards zero.