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This article explores the concepts of inverse relations and functions, detailing how to obtain a relation's inverse by switching the x and y coordinates in its equation. We describe the crucial relationship between the domain and range of a function and its inverse, emphasizing the graphical reflection over the line y=x. We also examine one-to-one functions and how certain tests (vertical and horizontal line tests) determine if a function is a one-to-one function. Additionally, we touch on exponential and logarithmic functions and their inverses.
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1-6 Inverses
What are the domain and range of Domain: Range: Interior of circle: Exterior of circle:
Inverse Relations and Functions To get a relation’s inverse, we switch x and y in the equation. If R is a relation, R-1 is the inverse of R. For every point (a, b) in R, (b, a) is in R-1. The domain of R-1 is the range of R and the range of R-1 is the domain of R. The graph of R-1 is a reflection of the graph of R over the line y = x.
Inverses that ARE functions If R-1 passes the vertical line test, then it is a function. Thus, if R passes the HORIZONTAL line test, then R-1 is a function. If R and R-1 are BOTH functions, then R is called a one-to-one function. (R passes both the VLT and the HLT.) If f(x) is one-to-one, then its inverse is f-1(x)
Inverse Functions (A function and its inverse “undo” each other.) Are f(x) and g(x) inverse functions? If not, what is f-1(x)?
Exponential and Logarithmic Functions is an exponential function with base a (a > 0) Domain: Range: y-intercept: a > 1 0 < a < 1
Exponential and Logarithmic Functions The inverse of is The inverse of is Domain: Range: x-intercept:
