Trigonometry Examples

Find the Inverse 4/49-(4/(7x))÷(x/49)-1/x
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Factor each term.
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Step 2.2.1
To divide by a fraction, multiply by its reciprocal.
Step 2.2.2
Cancel the common factor of .
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Step 2.2.2.1
Factor out of .
Step 2.2.2.2
Factor out of .
Step 2.2.2.3
Cancel the common factor.
Step 2.2.2.4
Rewrite the expression.
Step 2.2.3
Multiply by .
Step 2.2.4
Multiply by .
Step 2.2.5
Raise to the power of .
Step 2.2.6
Raise to the power of .
Step 2.2.7
Use the power rule to combine exponents.
Step 2.2.8
Add and .
Step 2.3
Find the LCD of the terms in the equation.
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Step 2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.3.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 2.3.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 2.3.4
has factors of and .
Step 2.3.5
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.3.6
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.3.7
Multiply by .
Step 2.3.8
The factors for are , which is multiplied by each other times.
occurs times.
Step 2.3.9
The factor for is itself.
occurs time.
Step 2.3.10
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.3.11
Multiply by .
Step 2.3.12
The LCM for is the numeric part multiplied by the variable part.
Step 2.4
Multiply each term in by to eliminate the fractions.
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Step 2.4.1
Multiply each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Simplify each term.
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Step 2.4.2.1.1
Cancel the common factor of .
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Step 2.4.2.1.1.1
Factor out of .
Step 2.4.2.1.1.2
Cancel the common factor.
Step 2.4.2.1.1.3
Rewrite the expression.
Step 2.4.2.1.2
Cancel the common factor of .
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Step 2.4.2.1.2.1
Move the leading negative in into the numerator.
Step 2.4.2.1.2.2
Factor out of .
Step 2.4.2.1.2.3
Cancel the common factor.
Step 2.4.2.1.2.4
Rewrite the expression.
Step 2.4.2.1.3
Multiply by .
Step 2.4.2.1.4
Cancel the common factor of .
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Step 2.4.2.1.4.1
Move the leading negative in into the numerator.
Step 2.4.2.1.4.2
Factor out of .
Step 2.4.2.1.4.3
Cancel the common factor.
Step 2.4.2.1.4.4
Rewrite the expression.
Step 2.4.2.1.5
Multiply by .
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Rewrite using the commutative property of multiplication.
Step 2.5
Solve the equation.
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Step 2.5.1
Subtract from both sides of the equation.
Step 2.5.2
Use the quadratic formula to find the solutions.
Step 2.5.3
Substitute the values , , and into the quadratic formula and solve for .
Step 2.5.4
Simplify the numerator.
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Step 2.5.4.1
Raise to the power of .
Step 2.5.4.2
Apply the distributive property.
Step 2.5.4.3
Multiply by .
Step 2.5.4.4
Multiply by .
Step 2.5.4.5
Apply the distributive property.
Step 2.5.4.6
Multiply by .
Step 2.5.4.7
Multiply by .
Step 2.5.4.8
Add and .
Step 2.5.4.9
Factor out of .
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Step 2.5.4.9.1
Factor out of .
Step 2.5.4.9.2
Factor out of .
Step 2.5.4.9.3
Factor out of .
Step 2.5.4.10
Rewrite as .
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Step 2.5.4.10.1
Factor out of .
Step 2.5.4.10.2
Rewrite as .
Step 2.5.4.10.3
Add parentheses.
Step 2.5.4.11
Pull terms out from under the radical.
Step 2.5.5
Simplify the expression to solve for the portion of the .
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Step 2.5.5.1
Change the to .
Step 2.5.5.2
Factor out of .
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Step 2.5.5.2.1
Factor out of .
Step 2.5.5.2.2
Factor out of .
Step 2.5.6
Simplify the expression to solve for the portion of the .
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Step 2.5.6.1
Simplify the numerator.
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Step 2.5.6.1.1
Raise to the power of .
Step 2.5.6.1.2
Apply the distributive property.
Step 2.5.6.1.3
Multiply by .
Step 2.5.6.1.4
Multiply by .
Step 2.5.6.1.5
Apply the distributive property.
Step 2.5.6.1.6
Multiply by .
Step 2.5.6.1.7
Multiply by .
Step 2.5.6.1.8
Add and .
Step 2.5.6.1.9
Factor out of .
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Step 2.5.6.1.9.1
Factor out of .
Step 2.5.6.1.9.2
Factor out of .
Step 2.5.6.1.9.3
Factor out of .
Step 2.5.6.1.10
Rewrite as .
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Step 2.5.6.1.10.1
Factor out of .
Step 2.5.6.1.10.2
Rewrite as .
Step 2.5.6.1.10.3
Add parentheses.
Step 2.5.6.1.11
Pull terms out from under the radical.
Step 2.5.6.2
Change the to .
Step 2.5.6.3
Factor out of .
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Step 2.5.6.3.1
Factor out of .
Step 2.5.6.3.2
Factor out of .
Step 2.5.6.3.3
Factor out of .
Step 2.5.7
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 .
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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 .
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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 .
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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 .
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Step 4.3.2.1
Divide each term in by and simplify.
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Step 4.3.2.1.1
Divide each term in by .
Step 4.3.2.1.2
Simplify the left side.
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Step 4.3.2.1.2.1
Cancel the common factor of .
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Step 4.3.2.1.2.1.1
Cancel the common factor.
Step 4.3.2.1.2.1.2
Divide by .
Step 4.3.2.1.3
Simplify the right side.
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Step 4.3.2.1.3.1
Divide by .
Step 4.3.2.2
Subtract from both sides of the inequality.
Step 4.3.2.3
Divide each term in by and simplify.
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Step 4.3.2.3.1
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 4.3.2.3.2
Simplify the left side.
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Step 4.3.2.3.2.1
Cancel the common factor of .
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Step 4.3.2.3.2.1.1
Cancel the common factor.
Step 4.3.2.3.2.1.2
Divide by .
Step 4.3.2.3.3
Simplify the right side.
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Step 4.3.2.3.3.1
Dividing two negative values results in a positive value.
Step 4.3.3
Set the denominator in equal to to find where the expression is undefined.
Step 4.3.4
Solve for .
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Step 4.3.4.1
Divide each term in by and simplify.
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Step 4.3.4.1.1
Divide each term in by .
Step 4.3.4.1.2
Simplify the left side.
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Step 4.3.4.1.2.1
Cancel the common factor of .
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Step 4.3.4.1.2.1.1
Cancel the common factor.
Step 4.3.4.1.2.1.2
Divide by .
Step 4.3.4.1.3
Simplify the right side.
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Step 4.3.4.1.3.1
Divide by .
Step 4.3.4.2
Subtract from both sides of the equation.
Step 4.3.4.3
Divide each term in by and simplify.
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Step 4.3.4.3.1
Divide each term in by .
Step 4.3.4.3.2
Simplify the left side.
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Step 4.3.4.3.2.1
Cancel the common factor of .
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Step 4.3.4.3.2.1.1
Cancel the common factor.
Step 4.3.4.3.2.1.2
Divide by .
Step 4.3.4.3.3
Simplify the right side.
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Step 4.3.4.3.3.1
Dividing two negative values results in a positive value.
Step 4.3.5
The domain is all values of that make the expression defined.
Step 4.4
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 5