which of the following are continuous for all real values of x?
I. f(x)= (x^2+5)/x^2+1
II. g(x)= 3/x^2
III. h(x)=|x|
a) I only
b) II only
c) I and II only
d) I and III only
Let's analyze each option one by one:
I. f(x) = (x^2+5)/(x^2+1)
Well, x^2+1 can never be equal to zero for any real value of x. We know that we cannot divide by zero, so the function is continuous for all real values of x.
II. g(x) = 3/x^2
Again, we can see that g(x) will not be defined if x equals zero, because we would be dividing by zero. So, g(x) is not continuous for all real values of x.
III. h(x) = |x|
The absolute value function is defined as the distance from zero on the number line. It is continuous everywhere except at x = 0, where there is a sharp turn in the graph. So, h(x) is not continuous for all real values of x.
Therefore, the only function that is continuous for all real values of x is option a) I only.
Hope that brightened up your day!
To determine which of the functions are continuous for all real values of x, we need to analyze each function individually.
I. f(x) = (x^2 + 5)/(x^2 + 1)
For this function, the only value that causes a discontinuity is when the denominator, x^2 + 1, equals zero. However, this never happens for any real value of x, since x^2 + 1 is always positive. Therefore, f(x) is continuous for all real values of x.
II. g(x) = 3/x^2
This function has a discontinuity when the denominator, x^2, equals zero. This occurs at x = 0. Therefore, g(x) is NOT continuous at x = 0.
III. h(x) = |x|
The absolute value function is continuous for all real values of x. Therefore, h(x) is continuous for all real values of x.
In summary:
I. f(x) is continuous for all real values of x.
II. g(x) is NOT continuous at x = 0.
III. h(x) is continuous for all real values of x.
The correct answer is:
d) I and III only
To determine which of the given functions are continuous for all real values of x, we need to check if there are any potential points of discontinuity.
I. f(x) = (x^2+5)/(x^2+1)
The function f(x) involves rational expressions. Rational functions are continuous for all x except for values that make the denominator zero. So, in this case, we need to check if x^2 + 1 = 0 has any real solutions.
x^2 + 1 = 0 implies x^2 = -1, which has no real solutions. Therefore, the denominator x^2 + 1 is never equal to zero, and thus f(x) is continuous for all real values of x.
II. g(x) = 3/x^2
Similar to the first function, g(x) is a rational function. To determine if g(x) is continuous for all x, we need to check if x^2 = 0 has any real solutions.
x^2 = 0 only has a single solution, x = 0. Since the function g(x) has a discontinuity at x = 0, it is NOT continuous for all real values of x.
III. h(x) = |x|
The function h(x) is the absolute value function. The absolute value function is continuous for all real values of x. So, h(x) is continuous for all real values of x.
Based on the analysis for each given function, the correct answer is:
d) I and III only
the function will be continuous if it does not have a denominator of zero
So, which of those fits that criterion?