Question

How to find the domain and range of the functions with all the steps.

And

Find the domain and range of the function f(x)=(9-x)^1/2

Answer 1

Step 1: See if you can determine what type of function you have (this isn’t always clear). Certain functions have defined domains and range.

• Linear functions and polynomial functions have the domain and range of all real numbers
• Square (quadratic) functions and absolute value functions have a domain of all real numbers and a range of y≥0
• Square root functions have a domain of x≥0 and a range of y≥0
• Rational functions have a domain of x≠0 and a range of x≠0.
• Sine functions and cosine functions have a domain of all real numbers and a range of -1 ≤y≥ 1
• Step 2: Try to use the guess and check method to determine the domain first. You’ll need a set of strong algebra skills to perform this step. For example, if you have the function f(x)=1/(x² – 9), you can exclude any values of x (the domain) that make the denominator equal to zero (because division by zero is not defined)

Step 3: Graph your function and see where your x-values and y-values lie. Most graphing calculators will help you see a function’s domain (or indicate which values might not be permissible. For example, if you graphed x^2, it would be clear that the domain cannot include negative numbers. IF you don’t have a graphing calculator.

Step 4: Make a table of values Include inputs of x from -10 to 10, then some larger numbers (like one million). Use a calculator to find values of y for values of x. If the calculator tells you the values or undefined, or that the values might be reaching a limit (a number that a function approaches, but never reaches), that should help you determine the range.

Solve to find the values of x that make the expression defined.

9−x2≥0

Move 9

to the right side of the equation by subtracting 9

from both sides of the equation.

−x2≥−9

Multiply each term in −x2≥−9

by −1

Tap for more steps...

x2≤9

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

x≤±√9

The complete solution is the result of both the positive and negative portions of the solution.

−3≤x≤3

Find the set of inequalities that expresses where x

is defined.

9−x2≥0

The domain is all values of x

that make the expression defined.

[−3,3]

{x|−3≤x≤3}

The range is the set of all valid y

values.

[0,3]{y|0≤y≤3}

Determine the domain and range.

Domain: [−3,3],{x|−3≤x≤3}

Range: [0,3],{y|0≤y≤3}