How to determine the new domain and range given the old domain and range?

"A function f(x) has domain $\{x \in \mathbb{R}\;|\;x \ge -4\}$ and range $\{y \in \mathbb{R}\;|\;y \lt -1\}$ Determine the domain and range of each function without graphing."

I was given the $2$ functions $y=f(-x)$ and $y=-2f(-x+5) + 1$. For the first function, I understand why the the domain is $x \lt 4$ (everything is flipped because of $-1$), but I don't understand why it doesn't say that $x \le 4$.

For the second function, I don't understand why the range doesn't say $y$ $\gt 3$ . I also don't understand why the domain isn't $x \le -1$. Hopefully this post made sense.

• Please check the first sentence in the last paragraph: "is less is greater than $1$" does not make sense. Also, please read this tutorial on how to typeset mathematics on this site. Using the proper mathematical symbolism would make your post easier to read. Commented Oct 2, 2016 at 21:51
• I don't understand how the LaTeX works. I typed y \ge 3 but it didn't work. Commented Oct 2, 2016 at 22:39
• See MathJax basic tutorial and quick reference. You need to enclose it in $ symbols, like $y \ge 3$. – dxiv Commented Oct 2, 2016 at 22:44 1 Answer For the second function, let$g(x) = -2 f(-x+5) + 1$. The argument passed to$f$is$-x + 5$, and it must be in the domain of$f$, so: $$-x + 5 \ge -4 \quad \iff \quad x \le 9$$ The range of values for$g(x)$follows from: $$f(-x + 5) \lt -1$$ $$-2 f(-x + 5) \gt 2$$ $$g(x) = -2 f(-x + 5) + 1 \gt 3$$ From the above, the domain of$g$is$\{x \in \mathbb{R}\;|\; x \le 9\}$and its range is$\{y \in \mathbb{R}\;|\; y \gt 3\}$. The first function can be worked out the same way, and it's obviously simpler. •$\{x \in \mathbb{R}\;|\;x \lt -1\}$and$\{y \in \mathbb{R}\;|\;y \lt 3\}$was the textbook's answer. Would you say the textbook's answer is incorrect? Commented Oct 2, 2016 at 22:53 • @Hamze That's either wrong, or maybe refers to a problem other than the one you posted. Simply by inspection it's easy to see that$f$is always negative, so$-2 f(-x+5)$is always positive, therefore the range of$y = -2 f(-x+5) + 1$can not include any negative numbers. – dxiv Commented Oct 2, 2016 at 23:01 • I don't understand, how do you know f is always negative? Commented Oct 2, 2016 at 23:06 • The domain is$x \ge -4$and the range is$y \lt -1$so$f(x) < -1 \lt 0$for$\forall x \ge -4$. – dxiv Commented Oct 2, 2016 at 23:08 • From the answer: "The argument passed to$f$is$−x+5$, and it must be in the domain of$f$". Stated another way,$f( \cdot )$is defined for$\cdot \ge -4$. Replace the$\cdot$with$-x+5\$ and the inequality follows as written.
– dxiv
Commented Oct 2, 2016 at 23:52