Finite Math Examples

Find the Inverse f(x)=x(-4x+180)
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
Tap for more steps...
Step 3.1
Rewrite the equation as .
Step 3.2
Simplify .
Tap for more steps...
Step 3.2.1
Simplify by multiplying through.
Tap for more steps...
Step 3.2.1.1
Apply the distributive property.
Step 3.2.1.2
Reorder.
Tap for more steps...
Step 3.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 3.2.1.2.2
Move to the left of .
Step 3.2.2
Multiply by by adding the exponents.
Tap for more steps...
Step 3.2.2.1
Move .
Step 3.2.2.2
Multiply by .
Step 3.3
Subtract from both sides of the equation.
Step 3.4
Use the quadratic formula to find the solutions.
Step 3.5
Substitute the values , , and into the quadratic formula and solve for .
Step 3.6
Simplify.
Tap for more steps...
Step 3.6.1
Simplify the numerator.
Tap for more steps...
Step 3.6.1.1
Raise to the power of .
Step 3.6.1.2
Multiply .
Tap for more steps...
Step 3.6.1.2.1
Multiply by .
Step 3.6.1.2.2
Multiply by .
Step 3.6.1.3
Factor out of .
Tap for more steps...
Step 3.6.1.3.1
Factor out of .
Step 3.6.1.3.2
Factor out of .
Step 3.6.1.3.3
Factor out of .
Step 3.6.1.4
Rewrite as .
Tap for more steps...
Step 3.6.1.4.1
Rewrite as .
Step 3.6.1.4.2
Rewrite as .
Step 3.6.1.5
Pull terms out from under the radical.
Step 3.6.1.6
Raise to the power of .
Step 3.6.2
Multiply by .
Step 3.6.3
Simplify .
Step 3.7
Simplify the expression to solve for the portion of the .
Tap for more steps...
Step 3.7.1
Simplify the numerator.
Tap for more steps...
Step 3.7.1.1
Raise to the power of .
Step 3.7.1.2
Multiply .
Tap for more steps...
Step 3.7.1.2.1
Multiply by .
Step 3.7.1.2.2
Multiply by .
Step 3.7.1.3
Factor out of .
Tap for more steps...
Step 3.7.1.3.1
Factor out of .
Step 3.7.1.3.2
Factor out of .
Step 3.7.1.3.3
Factor out of .
Step 3.7.1.4
Rewrite as .
Tap for more steps...
Step 3.7.1.4.1
Rewrite as .
Step 3.7.1.4.2
Rewrite as .
Step 3.7.1.5
Pull terms out from under the radical.
Step 3.7.1.6
Raise to the power of .
Step 3.7.2
Multiply by .
Step 3.7.3
Simplify .
Step 3.7.4
Change the to .
Step 3.8
Simplify the expression to solve for the portion of the .
Tap for more steps...
Step 3.8.1
Simplify the numerator.
Tap for more steps...
Step 3.8.1.1
Raise to the power of .
Step 3.8.1.2
Multiply .
Tap for more steps...
Step 3.8.1.2.1
Multiply by .
Step 3.8.1.2.2
Multiply by .
Step 3.8.1.3
Factor out of .
Tap for more steps...
Step 3.8.1.3.1
Factor out of .
Step 3.8.1.3.2
Factor out of .
Step 3.8.1.3.3
Factor out of .
Step 3.8.1.4
Rewrite as .
Tap for more steps...
Step 3.8.1.4.1
Rewrite as .
Step 3.8.1.4.2
Rewrite as .
Step 3.8.1.5
Pull terms out from under the radical.
Step 3.8.1.6
Raise to the power of .
Step 3.8.2
Multiply by .
Step 3.8.3
Simplify .
Step 3.8.4
Change the to .
Step 3.9
The final answer is the combination of both solutions.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
Tap for more steps...
Step 5.1
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 5.2
Find the range of .
Tap for more steps...
Step 5.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 5.3
Find the domain of .
Tap for more steps...
Step 5.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5.3.2
Solve for .
Tap for more steps...
Step 5.3.2.1
Subtract from both sides of the inequality.
Step 5.3.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 5.3.2.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 5.3.2.2.2
Simplify the left side.
Tap for more steps...
Step 5.3.2.2.2.1
Dividing two negative values results in a positive value.
Step 5.3.2.2.2.2
Divide by .
Step 5.3.2.2.3
Simplify the right side.
Tap for more steps...
Step 5.3.2.2.3.1
Divide by .
Step 5.3.3
The domain is all values of that make the expression defined.
Step 5.4
Find the domain of .
Tap for more steps...
Step 5.4.1
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 5.5
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 6