Restrictions are important when it comes to understanding the domain and range of functions. These ideas can be a little tricky, but they are really helpful.
Domain Restrictions:
Denominators: When you have a function that includes a fraction, like ( f(x) = \frac{1}{x-2} ), you need to be careful. If ( x ) equals 2, the function doesn’t work because you can’t divide by zero. So, the domain, which is all the possible values for ( x ), is everything except 2. You can write it like this: ( (-\infty, 2) \cup (2, \infty) ).
Square Roots: For a function like ( g(x) = \sqrt{x-3} ), you need to make sure that what’s inside the square root is zero or positive. This means ( x ) must be 3 or bigger. So, the domain here is ( [3, \infty) ).
Range Restrictions:
Output Values: The range is about the possible outputs of the function. For ( h(x) = \sqrt{x-3} ), the results will always be zero or positive because square roots can’t be negative. Therefore, the range is ( [0, \infty) ).
Behavior: If a function has certain special features, like horizontal or vertical lines it approaches but never touches, these can also change what the range can be.
Knowing these restrictions is super helpful. They make it easier to draw graphs and understand how functions work!
Restrictions are important when it comes to understanding the domain and range of functions. These ideas can be a little tricky, but they are really helpful.
Domain Restrictions:
Denominators: When you have a function that includes a fraction, like ( f(x) = \frac{1}{x-2} ), you need to be careful. If ( x ) equals 2, the function doesn’t work because you can’t divide by zero. So, the domain, which is all the possible values for ( x ), is everything except 2. You can write it like this: ( (-\infty, 2) \cup (2, \infty) ).
Square Roots: For a function like ( g(x) = \sqrt{x-3} ), you need to make sure that what’s inside the square root is zero or positive. This means ( x ) must be 3 or bigger. So, the domain here is ( [3, \infty) ).
Range Restrictions:
Output Values: The range is about the possible outputs of the function. For ( h(x) = \sqrt{x-3} ), the results will always be zero or positive because square roots can’t be negative. Therefore, the range is ( [0, \infty) ).
Behavior: If a function has certain special features, like horizontal or vertical lines it approaches but never touches, these can also change what the range can be.
Knowing these restrictions is super helpful. They make it easier to draw graphs and understand how functions work!