Trigonometry Examples

Find the Inverse k(x) = square root of 2x^2+5
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Rewrite the equation as .
Step 3.2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3.3
Simplify each side of the equation.
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Step 3.3.1
Use to rewrite as .
Step 3.3.2
Simplify the left side.
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Step 3.3.2.1
Simplify .
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Step 3.3.2.1.1
Multiply the exponents in .
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Step 3.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.3.2.1.1.2
Cancel the common factor of .
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Step 3.3.2.1.1.2.1
Cancel the common factor.
Step 3.3.2.1.1.2.2
Rewrite the expression.
Step 3.3.2.1.2
Simplify.
Step 3.4
Solve for .
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Step 3.4.1
Subtract from both sides of the equation.
Step 3.4.2
Divide each term in by and simplify.
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Step 3.4.2.1
Divide each term in by .
Step 3.4.2.2
Simplify the left side.
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Step 3.4.2.2.1
Cancel the common factor of .
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Step 3.4.2.2.1.1
Cancel the common factor.
Step 3.4.2.2.1.2
Divide by .
Step 3.4.2.3
Simplify the right side.
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Step 3.4.2.3.1
Move the negative in front of the fraction.
Step 3.4.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4.4
Simplify .
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Step 3.4.4.1
Combine the numerators over the common denominator.
Step 3.4.4.2
Rewrite as .
Step 3.4.4.3
Multiply by .
Step 3.4.4.4
Combine and simplify the denominator.
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Step 3.4.4.4.1
Multiply by .
Step 3.4.4.4.2
Raise to the power of .
Step 3.4.4.4.3
Raise to the power of .
Step 3.4.4.4.4
Use the power rule to combine exponents.
Step 3.4.4.4.5
Add and .
Step 3.4.4.4.6
Rewrite as .
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Step 3.4.4.4.6.1
Use to rewrite as .
Step 3.4.4.4.6.2
Apply the power rule and multiply exponents, .
Step 3.4.4.4.6.3
Combine and .
Step 3.4.4.4.6.4
Cancel the common factor of .
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Step 3.4.4.4.6.4.1
Cancel the common factor.
Step 3.4.4.4.6.4.2
Rewrite the expression.
Step 3.4.4.4.6.5
Evaluate the exponent.
Step 3.4.4.5
Combine using the product rule for radicals.
Step 3.4.4.6
Reorder factors in .
Step 3.4.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.4.5.1
First, use the positive value of the to find the first solution.
Step 3.4.5.2
Next, use the negative value of the to find the second solution.
Step 3.4.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
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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 .
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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 the inverse.
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Step 5.3.1
Find the domain of .
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Step 5.3.1.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5.3.1.2
Solve for .
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Step 5.3.1.2.1
Simplify .
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Step 5.3.1.2.1.1
Apply the distributive property.
Step 5.3.1.2.1.2
Multiply by .
Step 5.3.1.2.2
Graph each side of the equation. The solution is the x-value of the point of intersection.
Step 5.3.1.2.3
Use each root to create test intervals.
Step 5.3.1.2.4
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 5.3.1.2.4.1
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1.2.4.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.1.2.4.1.2
Replace with in the original inequality.
Step 5.3.1.2.4.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.1.2.4.2
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1.2.4.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.1.2.4.2.2
Replace with in the original inequality.
Step 5.3.1.2.4.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 5.3.1.2.4.3
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1.2.4.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.1.2.4.3.2
Replace with in the original inequality.
Step 5.3.1.2.4.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.1.2.4.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 5.3.1.2.5
The solution consists of all of the true intervals.
or
or
Step 5.3.1.3
The domain is all values of that make the expression defined.
Step 5.3.2
Find the union of .
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Step 5.3.2.1
The union consists of all of the elements that are contained in each interval.
Step 5.4
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 6