Evaluate the function at the indicated values. If the function does not exist at a value, enter NONE.
f(x)= |x| / x
f (-3) =
f (-1) =
f (0) =
f (3) =
f (x^2) =
f (1/x) =
|x|/x = 1 if x>0
|x|/x = -x/x = -1 if x<0
watch out for x=0
To evaluate the function at the indicated values, we substitute the values into the given function f(x) = |x| / x.
1. f(-3) = |-3| / -3 = 3 / -3 = -1
2. f(-1) = |-1| / -1 = 1 / -1 = -1
3. f(0) is undefined because division by zero is not defined in mathematics. Therefore, the function does not exist at x = 0.
4. f(3) = |3| / 3 = 3 / 3 = 1
5. f(x^2) = |x^2| / x^2
Since the absolute value of a square is always positive, we can simplify this equation to:
f(x^2) = x^2 / x^2 = 1
6. f(1/x) = |1/x| / (1/x)
When x ≠ 0, the absolute value of 1/x is always positive. So we can simplify this further:
f(1/x) = 1/x / (1/x) = 1
Therefore, the evaluated values are:
f(-3) = -1
f(-1) = -1
f(0) = NONE
f(3) = 1
f(x^2) = 1
f(1/x) = 1
To evaluate the function at the indicated values, we substitute the values into the function and simplify.
1. f(-3):
Substitute -3 into the function:
f(-3) = |-3| / -3
Simplify the expression:
f(-3) = 3 / -3
Finally, simplify the fraction:
f(-3) = -1
2. f(-1):
Substitute -1 into the function:
f(-1) = |-1| / -1
Simplify the expression:
f(-1) = 1 / -1
Finally, simplify the fraction:
f(-1) = -1
3. f(0):
Substitute 0 into the function:
f(0) = |0| / 0
The function does not exist when the denominator is zero, so the answer is NONE.
4. f(3):
Substitute 3 into the function:
f(3) = |3| / 3
Simplify the expression:
f(3) = 3 / 3
Finally, simplify the fraction:
f(3) = 1
5. f(x^2):
Substitute x^2 into the function:
f(x^2) = |x^2| / x^2
Simplify the expression:
f(x^2) = x^2 / x^2
Finally, simplify the fraction:
f(x^2) = 1
6. f(1/x):
Substitute 1/x into the function:
f(1/x) = |1/x| / (1/x)
Simplify the expression:
f(1/x) = |1| / (1/x)
Since |1| equals 1 and dividing by x is the same as multiplying by 1/x, we have:
f(1/x) = 1 / (1/x)
Simplify the expression:
f(1/x) = x / 1
Finally, simplify the fraction:
f(1/x) = x
In summary:
f(-3) = -1
f(-1) = -1
f(0) = NONE
f(3) = 1
f(x^2) = 1
f(1/x) = x