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1-7 Inverse Relations and Functions (Examples: (Find Inverse Functions…
1-7
Inverse Relations and Functions
Objectives
:
1-
Use the horizontal line test to determine inverse functions.
2-
Find inverse functions algebraically and graphically.
Vocabulary
Inverse Function:
If the inverse relation of a function f is also a function, then it is called the inverse function of f and is denoted f-1, read f inverse.
One To One:
If a function passes the horizontal line test, then it is said to be one-to-one, because no x-value is matched with more than one y-value and no y-value is matched with more than one x-value.
Inverse Relations:
Exist if and only if one relation contains (b, a) whenever the other relation contains (a, b).
Examples:
Find Inverse Functions Algebraically
f(x)=√x - 4
The graph of f shown passes the horizontal line test. Therefore, f is a one-to-one function and has an inverse function. From the graph, you can see that f has domain [4, ∞) and range [0, ∞). Now find f-1.
f ( x ) = √ x - 4
y = √ x - 4
x = √ y - 4 x^2 = y - 4
y=x^2+4 f^-1(x)=x^2+4
f(x)=√x - 4
f^-1(x)=x^2+4
Verify Inverse Functions
Because f[g(x)] = g[ f(x)] = x, f(x) and g(x) are inverse functions. This is supported graphically because f(x) and g(x) appear to be reflections of each other in the line y = x.
Apply the Horizontal Line Test
f(x)=|x-1|
The graph of f(x) in Figure 1.7.1 shows that it is possible to find a horizontal line that intersects the graph of f(x) more than once. Therefore, you can conclude that f-1 does not exist.
Find Inverse Functions Graphically
Use the graph of f (x) in Figure 1.7.3 to graph f ^-1(x).
Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x (Figure 1.7.4).
Relation
Inverse Relation
Horizontal Line Test:
A function f has an inverse function f -1 if and only if each horizontal line intersects the graph of the function in at most one point.
