Given the function f(x)=4|x|, write f(x) as a piecewise function that does not include absolute value symbols.Please fill the blanks
f(x)=blank; when x≤ blank
f(x)= blank; when x> blank
f(x) = 4x when x ≤ 0
f(x) = 4 x when x > 0
| x | is always positive so 4 | x | is also always positive.
4 | x | = 4 x for all value of variable x.
f(x) = -4x when x ≤ 0
|x| = x if x >= 0
|x| = -x if x < 0
To rewrite the function f(x) = 4|x| without absolute value symbols, we need to consider the different cases when x is positive or negative.
When x ≤ 0, since |x| = -x when x ≤ 0, we have:
f(x) = 4|x| = 4(-x) = -4x
Therefore, when x ≤ 0, f(x) = -4x.
When x > 0, |x| = x when x > 0, so we have:
f(x) = 4|x| = 4(x) = 4x
Therefore, when x > 0, f(x) = 4x.
Combining both cases, we can write f(x) as a piecewise function:
f(x) = -4x; when x ≤ 0
f(x) = 4x; when x > 0
To write the function f(x) = 4|x| as a piecewise function without absolute value symbols, we need to consider the different cases when x is positive or negative. Let's fill in the blanks:
1. f(x) = -4x; when x ≤ 0
- When x is less than or equal to zero, the absolute value of x is simply -x. Hence, we substitute -4x in place of |x|, as the negative sign ensures the value is negative.
2. f(x) = 4x; when x > 0
- When x is greater than zero, the absolute value of x is x itself. Therefore, we can directly substitute 4x in place of |x|.
So the piecewise function representation for f(x) = 4|x| without absolute value symbols is:
f(x) = -4x; when x ≤ 0
f(x) = 4x; when x > 0