4.1 Inverses Mon March 23 Do Now Solve for Y 1) 2)
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Transcript of 4.1 Inverses Mon March 23 Do Now Solve for Y 1) 2)
Inverses
• When we go from an output of a function back to its inputs, we get an inverse relation
• Interchanging the first and second coordinates of each ordered pair produces the inverse relation
One-to-one Functions
• A function f is one-to-one if different inputs have different outputs
• If
• Every y-value is unique
Properties of one-to-one functions
• If a function f is one-to-one, its inverse is a function
• The domain of a one-to-one function f is the range of its inverse
• The range of a one-to-one function is the domain of its inverse
• A function that is always increasing or decreasing is one-to-one
Ways to show a function is one-to-one
• 1) Assume f(a) = f(b); then show that a = b– If you can think of a y-value that has 2 x-values (ex:
x^2
• 2) Horizontal Line Test– If a horizontal line intersects the graph more than
once, it is NOT one-to-one
How to find a formula for inverse
• 1) Replace f(x) with y (if possible) • 2) Switch x and y• 3) Solve for y• 4) Replace y with
Inverses and Graphs
• The graph of an inverse function is a reflection of f(x)’s graph across the line y = x
Inverse Functions and Compositions
• If a function f(x) is one-to-one, then the following compositions are true:
for each x in the domain of f
for each x in the domain of f inverse
Restricting the domain
• In the case in which the inverse of a function is not a function, the domain can be restricted to allow the inverse to be a function
• This is why square root graphs only show half of the true graph in a calculator