Holt McDougal Algebra 2 Functions and Their Inverses Holt Algebra2Holt McDougal Algebra 2 How do we determine whether the inverse of a function is a function?
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Holt McDougal Algebra 2
Functions and Their InversesFunctions and Their Inverses
Holt Algebra2Holt McDougal Algebra 2
• How do we determine whether the inverse of a function is a function?
• How do we write rules for the inverses of functions?
Holt McDougal Algebra 2
Functions and Their Inverses
In previous lessons, you learned that the inverse of a function f(x) “undoes” f(x). Its graph is a reflection across line y = x. The inverse may or not be a function.
Recall that the vertical-line test can help you determine whether a relation is a function. Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.
Holt McDougal Algebra 2
Functions and Their Inverses
Holt McDougal Algebra 2
Functions and Their Inverses
1. Use the horizontal-line test to determine whether the inverse of the blue relation is a function.
Using the Horizontal-Line Test
The inverse is a function because no horizontal line passes through two points on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
2. Use the horizontal-line test to determine whether the inverse of the red relation is a function.
Using the Horizontal-Line Test
The inverse is a not a function because a horizontal line passes through more than one point on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
3. Use the horizontal-line test to determine whether the inverse of the red relation is a function.
Using the Horizontal-Line Test
The inverse is a function because no horizontal line passes through two points on the graph.
Holt McDougal Algebra 2
Functions and Their Inverses
Recall from previous lessons that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
4. Find the inverse of . Determine whether it is a function, and state its domain and range.
Step 1 Graph the function.
The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
4. Find the inverse of . Determine whether it is a function, and state its domain and range.
Rewrite the function using y instead of f(x).
Step 2 Find the inverse.
Simplify.
Switch x and y in the equation.
Cube both sides.
Isolate y.
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
4. Find the inverse of . Determine whether it is a function, and state its domain and range.Because the inverse is a function, .
The domain of the inverse is the range of f(x):{x|x R}.
The range is the domain of f(x):{y|y R}.Check Graph both relations to see that they are
symmetric about y = x.
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
5. Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
Step 1 Graph the function.
The horizontal-line test shows that the inverse is not a function. Note that the domain of f is all real numbers but the range is [ 4,
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
Rewrite the function using y instead of f(x).
Step 2 Find the inverse.
Take the square root of both sides.
Switch x and y in the equation.
Add 4 to both sides of the equation.
Simplify.
5. Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
y = x2 – 4
x = y2 – 4
x + 4 = y2
+ - x + 4 = y
2+ - x + 4 = y
Holt McDougal Algebra 2
Functions and Their InversesWriting Rules for inverses
5. Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
The domain of the inverse is the range of f(x): [ 4,
The range is the domain of f(x): R.Check Graph both relations to see that they
are symmetric about y = x.
Because the inverse is not a function, . 41 xxf
Holt McDougal Algebra 2
Functions and Their Inverses
Lesson 14.2 Practice A
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