Understanding Domain and Range of Functions

Figuring out the domain and range of different types of functions can feel a bit like solving a puzzle. But don’t worry! Once you learn some strategies, it gets easier. Let’s break it down.

Domain

The domain is all the possible input values (mostly $x$ values) that a function can use. Here’s how to find the domain for different kinds of functions:

1. Polynomial Functions:

   • These are usually simple! For example, in the function $f(x) = x^2 - 4x + 4$, you can use any real number for $x$. So, the domain is all real numbers.

2. Rational Functions:

   • This is where it gets a bit trickier. You need to avoid values of $x$ that would make the bottom part (denominator) zero. For example, with $g(x) = \frac{1}{x-3}$, the domain is all real numbers except $x = 3$.

3. Square Root Functions:

   • For functions like $h(x) = \sqrt{x - 1}$, you have to make sure everything inside the square root is zero or positive. Set $x - 1 \geq 0$, which leads to $x \geq 1$. So, the domain here is from $1$ to infinity, written as $[1, \infty)$.

4. Logarithmic Functions:

   • For a function like $j(x) = \log(x + 5)$, the number inside the log must be positive. Solve $x + 5 > 0$, which tells you $x > -5$. The domain is $(-5, \infty)$.

Range

The range is all the possible output values (mostly $y$ values) that the function can give. Here are some tips for finding the range:

1. Linear Functions:

   • For a function like $f(x) = 2x + 3$, the range is all real numbers because you can get any $y$ value by changing $x$.

2. Quadratic Functions:

   • If you have a quadratic like $f(x) = -x^2 + 4$, first find the vertex, which shows you the highest or lowest point. This one opens down, so the range is $(-\infty, 4]$.

3. Trigonometric Functions:

   • For functions like $k(x) = \sin(x)$, the range is always between $-1$ and $1$, no matter what you input!

4. Exponential Functions:

   • For a function like $m(x) = 2^x$, the range is $(0, \infty)$. It never actually reaches zero but can get very close.

With some practice, finding the domain and range will feel natural! Just remember to check for any limits based on the type of function you have.