How do you solve f(x)=x * |-1|^x?
f(x) = x * |-1|^x can be simplified as follows:
- If x is even, then |-1|^x = 1, so f(x) = x * 1 = x.
- If x is odd, then |-1|^x = -1, so f(x) = x * (-1) = -x.
Therefore, we can write the solution as:
f(x) = {
x, if x is even,
-x, if x is odd.
}
How do you solve Use the function rule f(x)=x * |-1|^x find the output f(2)
Using the function rule f(x) = x * |-1|^x, we can find the output f(2) by plugging in 2 for x:
f(2) = 2 * |-1|^2
Since the absolute value of -1 is 1, we can simplify this expression further:
f(2) = 2 * 1^2
And since any number raised to the power of 2 is equal to itself squared:
f(2) = 2 * 1
Thus, the output of f(2) is 2.
To solve the equation f(x) = x * |-1|^x, we need to find the value(s) of x that satisfy the equation.
Let's break it down step by step:
1. Start by analyzing the function |x|, which represents the absolute value of x. The absolute value of a number is its distance from zero on the number line, regardless of its sign. Therefore, |x| = x when x is positive or zero, and |x| = -x when x is negative.
2. Now look at the base of the exponential term: |-1|^x. Since -1 raised to any power is always -1, regardless of whether the power is even or odd, we can simplify |-1|^x as (-1)^x.
3. Combine the absolute value and exponentiation: f(x) = x * (-1)^x.
To find the solutions, we need to consider different cases based on the sign of x:
Case 1: x > 0
In this case, the absolute value of x is the same as x, so the equation becomes f(x) = x * (-1)^x.
To solve this equation, we need to evaluate (-1)^x. Since (-1) raised to an even power is 1, and (-1) raised to an odd power is -1, we have two scenarios:
- When x is even, (-1)^x = 1, so f(x) = x * 1 = x.
- When x is odd, (-1)^x = -1, so f(x) = x * (-1) = -x.
Therefore, when x > 0, we have two solutions: x and -x.
Case 2: x = 0
Since x is zero, the equation f(x) = x * (-1)^x reduces to f(0) = 0 * (-1)^0 = 0 * 1 = 0.
Therefore, x = 0 is a solution.
Case 3: x < 0
In this case, the absolute value of x is -x, so the equation becomes f(x) = x * (-1)^x.
Similar to case 1, the evaluation of (-1)^x depends on the parity of x:
- When x is even, (-1)^x = 1, so f(x) = x * 1 = x.
- When x is odd, (-1)^x = -1, so f(x) = x * (-1) = -x.
Again, we have two solutions: x and -x.
To summarize, the solutions to the equation f(x) = x * |-1|^x are:
- When x > 0, the solutions are x and -x.
- When x = 0, the solution is 0.
- When x < 0, the solutions are x and -x.