Trigonometry Examples

Find the Inverse (x^2)/8+(3x^2)/2+(9x)/2
Step 1
Interchange the variables.
Step 2
Solve for .
Tap for more steps...
Step 2.1
Rewrite the equation as .
Step 2.2
Simplify .
Tap for more steps...
Step 2.2.1
To write as a fraction with a common denominator, multiply by .
Step 2.2.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps...
Step 2.2.2.1
Multiply by .
Step 2.2.2.2
Multiply by .
Step 2.2.3
Combine the numerators over the common denominator.
Step 2.2.4
Simplify each term.
Tap for more steps...
Step 2.2.4.1
Simplify the numerator.
Tap for more steps...
Step 2.2.4.1.1
Factor out of .
Tap for more steps...
Step 2.2.4.1.1.1
Multiply by .
Step 2.2.4.1.1.2
Factor out of .
Step 2.2.4.1.1.3
Factor out of .
Step 2.2.4.1.2
Multiply by .
Step 2.2.4.1.3
Add and .
Step 2.2.4.2
Move to the left of .
Step 2.3
Subtract from both sides of the equation.
Step 2.4
Multiply through by the least common denominator , then simplify.
Tap for more steps...
Step 2.4.1
Apply the distributive property.
Step 2.4.2
Simplify.
Tap for more steps...
Step 2.4.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Rewrite the expression.
Step 2.4.2.2
Cancel the common factor of .
Tap for more steps...
Step 2.4.2.2.1
Factor out of .
Step 2.4.2.2.2
Cancel the common factor.
Step 2.4.2.2.3
Rewrite the expression.
Step 2.4.2.3
Multiply by .
Step 2.4.2.4
Multiply by .
Step 2.4.3
Move .
Step 2.5
Use the quadratic formula to find the solutions.
Step 2.6
Substitute the values , , and into the quadratic formula and solve for .
Step 2.7
Simplify.
Tap for more steps...
Step 2.7.1
Simplify the numerator.
Tap for more steps...
Step 2.7.1.1
Raise to the power of .
Step 2.7.1.2
Multiply .
Tap for more steps...
Step 2.7.1.2.1
Multiply by .
Step 2.7.1.2.2
Multiply by .
Step 2.7.1.3
Factor out of .
Tap for more steps...
Step 2.7.1.3.1
Factor out of .
Step 2.7.1.3.2
Factor out of .
Step 2.7.1.3.3
Factor out of .
Step 2.7.1.4
Rewrite as .
Tap for more steps...
Step 2.7.1.4.1
Rewrite as .
Step 2.7.1.4.2
Rewrite as .
Step 2.7.1.5
Pull terms out from under the radical.
Step 2.7.1.6
Raise to the power of .
Step 2.7.2
Multiply by .
Step 2.7.3
Simplify .
Step 2.8
Simplify the expression to solve for the portion of the .
Tap for more steps...
Step 2.8.1
Simplify the numerator.
Tap for more steps...
Step 2.8.1.1
Raise to the power of .
Step 2.8.1.2
Multiply .
Tap for more steps...
Step 2.8.1.2.1
Multiply by .
Step 2.8.1.2.2
Multiply by .
Step 2.8.1.3
Factor out of .
Tap for more steps...
Step 2.8.1.3.1
Factor out of .
Step 2.8.1.3.2
Factor out of .
Step 2.8.1.3.3
Factor out of .
Step 2.8.1.4
Rewrite as .
Tap for more steps...
Step 2.8.1.4.1
Rewrite as .
Step 2.8.1.4.2
Rewrite as .
Step 2.8.1.5
Pull terms out from under the radical.
Step 2.8.1.6
Raise to the power of .
Step 2.8.2
Multiply by .
Step 2.8.3
Simplify .
Step 2.8.4
Change the to .
Step 2.8.5
Factor out of .
Tap for more steps...
Step 2.8.5.1
Factor out of .
Step 2.8.5.2
Factor out of .
Step 2.8.6
Rewrite as .
Step 2.8.7
Factor out of .
Step 2.8.8
Factor out of .
Step 2.8.9
Move the negative in front of the fraction.
Step 2.9
Simplify the expression to solve for the portion of the .
Tap for more steps...
Step 2.9.1
Simplify the numerator.
Tap for more steps...
Step 2.9.1.1
Raise to the power of .
Step 2.9.1.2
Multiply .
Tap for more steps...
Step 2.9.1.2.1
Multiply by .
Step 2.9.1.2.2
Multiply by .
Step 2.9.1.3
Factor out of .
Tap for more steps...
Step 2.9.1.3.1
Factor out of .
Step 2.9.1.3.2
Factor out of .
Step 2.9.1.3.3
Factor out of .
Step 2.9.1.4
Rewrite as .
Tap for more steps...
Step 2.9.1.4.1
Rewrite as .
Step 2.9.1.4.2
Rewrite as .
Step 2.9.1.5
Pull terms out from under the radical.
Step 2.9.1.6
Raise to the power of .
Step 2.9.2
Multiply by .
Step 2.9.3
Simplify .
Step 2.9.4
Change the to .
Step 2.9.5
Factor out of .
Tap for more steps...
Step 2.9.5.1
Reorder and .
Step 2.9.5.2
Factor out of .
Step 2.9.5.3
Factor out of .
Step 2.9.5.4
Factor out of .
Step 2.9.6
Move the negative in front of the fraction.
Step 2.10
The final answer is the combination of both solutions.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
Tap for more steps...
Step 4.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 4.2
Find the range of .
Tap for more steps...
Step 4.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 4.3
Find the domain of .
Tap for more steps...
Step 4.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 4.3.2
Solve for .
Tap for more steps...
Step 4.3.2.1
Subtract from both sides of the inequality.
Step 4.3.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 4.3.2.2.1
Divide each term in by .
Step 4.3.2.2.2
Simplify the left side.
Tap for more steps...
Step 4.3.2.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 4.3.2.2.2.1.1
Cancel the common factor.
Step 4.3.2.2.2.1.2
Divide by .
Step 4.3.2.2.3
Simplify the right side.
Tap for more steps...
Step 4.3.2.2.3.1
Move the negative in front of the fraction.
Step 4.3.3
The domain is all values of that make the expression defined.
Step 4.4
Find the domain of .
Tap for more steps...
Step 4.4.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 4.5
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 5