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Functions and Their Inverses. Essential Questions. 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. Holt Algebra2.
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Functions and Their Inverses Essential Questions • 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 Holt Algebra2
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.
Using the Horizontal-Line Test Use the horizontal-line test to determine whether the inverse of the blue relation is a function. The inverse is a function because no horizontal line passes through two points on the graph.
Using the Horizontal-Line Test Use the horizontal-line test to determine whether the inverse of the red relation is a function. The inverse is a not a function because a horizontal line passes through more than one point on the graph.
Using the Horizontal-Line Test Use the horizontal-line test to determine whether the inverse of the red relation is a function. The inverse is a function because no horizontal line passes through two points on the graph.
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.
Find the inverse of . Determine whether it is a function, and state its domain and range. Writing Rules for inverses Step 1 Graph the function. The horizontal-line test shows that theinverse is a function. Note that the domain and range of f are all real numbers.
Find the inverse of . Determine whether it is a function, and state its domain and range. Writing Rules for inverses Step 2 Find the inverse. Rewrite the function using y instead of f(x). Switch x and y in the equation. Cube both sides. Simplify. Isolate y.
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|xR}. The range is the domain of f(x):{y|yR}. Writing Rules for inverses CheckGraph both relations to see that they are symmetric about y = x.
Writing Rules for inverses 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 theinverse is not a function. Note that the domain of f is all real numbers but the range is [ -4, +µ).
2 + - x + 4 = y + - x + 4 = y Writing Rules for inverses Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range. Step 2 Find the inverse. Rewrite the function using y instead of f(x). y= x2 – 4 Switch x and y in the equation. x= y2 – 4 x + 4= y2 Add 4 to both sides of the equation. Take the square root of both sides. Simplify.
Writing Rules for inverses Find the inverse of f(x) = x2 – 4. Determine whether it is a function, and state its domain and range. Because the inverse is not a function, . The domain of the inverse is the range of f(x): [ -4, +µ). The range is the domain of f(x): R. CheckGraph both relations to see that they are symmetric about y = x.
