I'm 2 days into my AP calc class and I'm assigned a paper on basic algebraic functions and graphs of them (like domain and range, piecewise stuff), but I totally can't remember the stuff about it from algebra.
The instructions are "state the domain and range for the following. Find when f(x)=0. Graph the function"
My first problem is: f(x)=√3x+4+13
(the '+4' is also under the radical with the 3x)
Please tell me how to solve and answer this problem

Question

I'm 2 days into my AP calc class and I'm assigned a paper on basic algebraic functions and graphs of them (like domain and range, piecewise stuff), but I totally can't remember the stuff about it from algebra.
The instructions are "state the domain and range for the following. Find when f(x)=0. Graph the function" 
My first problem is: f(x)=√3x+4+13
(the '+4' is also under the radical with the 3x)
Please tell me how to solve and answer this problem

anonymous · Answer

To find the domain and range of a certain function, it helps to compare it to a similar, simpler function.

In this case, you're given $f(x)=\sqrt{3x+4}+13$, which is a lot like $g(x)=\sqrt x$.

Do you the domain/range of $g(x)$ ?

anonymous · Answer

Uh please rephrase your question I think you're missing a word

anonymous · Answer

*know*

Are you familiar with the domain/range of $\sqrt x$ ? Sorry for the confusion

anonymous · Answer

No I am not

anonymous · Answer

One way of saying it is, the domain of a function is the set of all numbers that can be used as "input" so that the "output" exists. The range is the set of all numbers that can possibly be the output.

In this case, for $\sqrt x$ to be real, you must have $x\ge 0$. So, the domain is all non-negative numbers, which can be expressed in a variety of ways. I'll stick with the interval notation, $x\ge 0$.

The range of $\sqrt x$ is also all non-negative numbers. If you use a positive real number as input, such as $x=4$, your output will also be positive, such as $\sqrt{4}=2$. The range is thus $g(x)\ge0$.

For $f(x)$, you have to consider the entire quantity under the radical. For $\sqrt{3x+4}+13$ to be defined, you have the restriction that $3x+4$ is non-negative, or
$$3x+4\ge0$$
Solving for $x$ gives you the domain of $f(x)$.

The range of $f(x)$ is similar to $g(x)$. Since $\sqrt{3x+4}$ gives you a number greater than or equal to 0, you have
$$\sqrt{3x+4}\ge0$$
which gives you
$$\sqrt{3x+4}+13\ge13~\iff~f(x)\ge13$$
This would be your range.

Does all this make sense?

anonymous · Answer

yes actually, thank you very much. I have some more questions though can you help me with the next one?

anonymous · Answer

Sure. Could you post it as another question? In case I can't help you, someone else might.

anonymous · Answer

yeah, I can answer the next few on my paper with the info you gave me in this one but i'll probably need help later