# Show that the following sum of PDF's is also a valid PDF

The specific question that I'm having trouble with is:

Let $$F$$ be the CDF of a continuous random variable, and let $$f = F^{'}$$.

Let function $$h(x) = \frac{1}{2}f(-x) + \frac{1}{2}f(x)$$. Show that $$h$$ is a valid PDF.

My instructor provided a solution as follows:

$$\int_{-\infty}^{\infty}h(x)dx = \frac{1}{2}\int_{-\infty}^{\infty}f(-x)dx\ +\ \frac{1}{2}\int_{\infty}^{\infty}f(x)dx\ =\ 1$$

I'm not sure how he derived this. I attempted to solve it via symmetry, but I'm not sure if it is applicable here, as there is no information about $$f(x)$$.

How did this result come to be?

Thank you.

Since $$f$$ is a PDF, it integrates to one (i.e., $$\int_{-\infty}^{\infty}f(x)dx=1$$) by definition.
Can you prove that $$\int_{-\infty}^{\infty} f(-x)dx=1$$ also?