for which of following functions is graphof y = |f(x)| identical to graph of y = f(x)?

A. f(x) = 2x
B. f(x) = -2x
C. f(x) = 2x^2
D. f(x) = -2x^2

i real not get this.

has to be positive for all x

has to be an even function, same + or -
so
2 x^2

ok can u explain to me again that i still not get it.

|f(x)| is positive, but

2x can be negative
-2x can be negative

x^2 is always positive, so

-2x^2 is always negative.

The only choice which is always positive is 2x^2

so, |2x^2| = 2x^2

To determine which of the given functions has a graph of |f(x)| identical to the graph of f(x), we need to understand what the absolute value function does to a graph.

The absolute value function, |x|, takes the input x and gives its non-negative value. In other words, it disregards the sign of x and gives the positive value of x, including 0.

Now, let's apply this concept to each function:

A. f(x) = 2x
To find |f(x)|, we take the absolute value of 2x. Since 2x can be both positive and negative, |f(x)| will have the same graph as f(x) only on the positive side of the x-axis. So, A is not the correct option.

B. f(x) = -2x
Similarly, taking the absolute value of -2x gives us |f(x)|. Again, |f(x)| will have the same graph as f(x) only on the negative side of the x-axis. Thus, B is not the correct option either.

C. f(x) = 2x^2
Here, we are squaring x before multiplying by 2. Since squaring a number always results in a non-negative value or 0, f(x) = 2x^2 will always yield non-negative values. Consequently, |f(x)| will be identical to f(x) for all values of x. Therefore, C is the correct option.

D. f(x) = -2x^2
When we square x and multiply by -2, f(x) = -2x^2 will give negative values unless x = 0. So, |f(x)| will not be identical to f(x) since it will always be positive or 0.

In conclusion, the function f(x) = 2x^2 is the only option for which the graph of y = |f(x)| is identical to the graph of y = f(x).