Graphs: Domain, Range, Functions, and Equation Restrictions

Hey friend! Let's talk about functions, domain, and range. Don't worry, it's not as scary as it sounds, it's actually pretty cool once you get the hang of it!

1. What's the "Domain"?

Think of a function like a little machine. You put something in (an input), and it spits something out (an output). The Domain is basically all the stuff you're allowed to put into the machine without breaking it or making it unhappy.

There are a few big "no-nos" you need to watch out for:

1. No Dividing by Zero! This is super important. If you have a fraction in your function and there's an $x$ in the bottom (the denominator), you cannot let that bottom part become zero. Why? Because dividing by zero breaks maths! So, if you see something like $f(x) = \frac{1}{x-3}$, you'd set that bottom part ($x-3$) to zero and say, "Nope, $x$ can't be 3 here!" So your domain would be all numbers except 3.

2. No Square-Rooting Negatives! (Or any even root, like a 4th root, 6th root, etc.). If you have a square root symbol $( \sqrt{} )$ in your function, whatever is inside that symbol must be zero or positive. It can't be a negative number, because you can't get a real number answer from that. So, for $h(x) = \sqrt{x+2}$, you'd say, "Whatever's under here ($x+2$) has to be greater than or equal to 0." That means $x$ has to be -2 or bigger.

3. Logs Need Positive Stuff Inside! (This is the Logarithmic Rule you asked about!). For any logarithm function, like $\,log(x)$, the stuff inside the parentheses (called the argument) must always be strictly positive. It can't be zero, and it can't be negative. So, if you have $f(x) = \log(x-1)$, you'd tell yourself, "Okay, $x-1$ has to be greater than 0." That means $x$ has to be bigger than 1. No zeros, no negatives inside those logs!

4. No Restrictions (Yay, easy!) If your function is just a plain old polynomial (like $f(x) = x^2 + 2x - 5$) with no fractions, no square roots, and no logs, then you can throw any number in there! The domain is all real numbers, from $-\infty$ to $+\infty$.

2. What's the "Range"?

Alright, so you know what you can put in (the domain). Now, the Range is all the stuff that actually comes out of the function. It's the set of all possible output values that the function can give you.

Figuring out the range can be a bit trickier sometimes, but here are some friendly tips:

1. Look at the Picture! (Graphing) This is often the easiest way. If you can draw your function, just look at how far up and down the graph goes. The lowest $y$-value it hits to the highest $y$-value it hits (or approaches) is your range.

2. Know Your Basic Shapes! Remember how a squaring function like $y=x^2$ always gives you results that are zero or positive? So its range is $[0, \infty)$. And for $y=|x|$, it's the same! Knowing these basic "parent functions" helps a lot.

3. Think about Shifts and Stretches! If you know that $y=x^2$ has outputs from 0 upwards, then $y=x^2+2$ will just take those outputs and add 2 to them, so now the range starts at 2 and goes upwards ($[2, \infty)$).

4. Sometimes, a Little Swap-a-Roo! For some functions, you can try to switch the $x$ and $y$ and solve for $y$ again. If you can find the domain of that new function, it'll tell you the range of your original function. It's like finding the input limits of the "reverse" machine to see what its possible outputs were!

Practice Problems:

Directions: Find the domain and range for each of the following functions.

1. $f(x) = 3x - 5$

   • Domain:

   • Range:

2. $g(x) = \frac{1}{x+4}$

   • Domain:

   • Range:

3. $h(x) = \sqrt{x-1}$

   • Domain:

   • Range:

4. $k(x) = x^2 + 2$

   • Domain:

   • Range:

5. $m(x) = |x| - 3$

   • Domain:

   • Range:

Solutions:

1. $f(x) = 3x - 5$

   • Domain: All real numbers, or $(-\infty, \infty)$. (It's a straight line, no weird stuff!)

   • Range: All real numbers, or $(-\infty, \infty)$. (A straight line goes up and down forever!)

2. $g(x) = \frac{1}{x+4}$

   • Domain: $x \neq -4$, or $(-\infty, -4) \cup (-4, \infty)$. (Can't divide by zero, so $x+4$ can't be 0!)

   • Range: $y \neq 0$, or $(-\infty, 0) \cup (0, \infty)$. (This function will get super close to zero but never actually hit it, because to get 0 out, the top would need to be 0, and it's 1!)

3. $h(x) = \sqrt{x-1}$

   • Domain: $x \ge 1$, or $[1, \infty)$. (What's inside the square root ($x-1$) has to be 0 or positive!)

   • Range: $y \ge 0$, or $[0, \infty)$. (Square roots always give you 0 or positive numbers as outputs!)

4. $k(x) = x^2 + 2$

   • Domain: All real numbers, or $(-\infty, \infty)$. (Just a simple parabola, no restrictions!)

   • Range: $y \ge 2$, or $[2, \infty)$. (An $x^2$ always gives results 0 or positive. If you add 2, the smallest output you can get is $0+2=2$.)

5. $m(x) = |x| - 3$

   • Domain: All real numbers, or $(-\infty, \infty)$. (You can take the absolute value of any number!)

   • Range: $y \ge -3$, or $[-3, \infty)$. (The absolute value $|x|$ always gives results 0 or positive. If you subtract 3, the smallest output you can get is $0-3=-3$. )