Inverse, Exponential, and Logarithmic Functions

Question 1

Inverse Function (definition)

Answer 1

A function f, its inverse (if it exists) is a function f^-1

y=f(x), then f^-1(y)=x

Question 2

Natural Exponential Function (definition)

Answer 2

f(x)=e^x —> e=2.178…..

Question 3

One-to-One Functions and the Horizontal Line Test

Answer 3

one-to-one: on a domain D is each value of f(x) corresponds to exactly one value of x in D.

horizontal line test: every horizontal line intersects the graph of a one-to-one function at most once.

Question 4

Existence of Inverse Functions

Answer 4

Let f be a one-to-one function on a domain D with a range R. Then f has a unique inverse f^-1 with domain R and range D such that f^-1(f(x))=x and f(f^-1(y))=y. Where x is in D and y is in R.

Question 5

Procedure Finding an Inverse Function

Answer 5

Suppose f is one-to-one on an interval I. To find f^-1 use the following steps.
1. Solve y=f(x) for x. If necessary, choose the function that corresponds to I.
2. Interchange x and y and write y=f^-1(x).

Question 6

Logarithmic Function Base b (definition)

Answer 6

for any base b>0, with b≠1, the logarithmic function base b,
denoted y=logbx, is the inverse of the exponential function y=b^x. base b=e is the natural logarithm function, denoted y=Inx.

Question 7

Change-of-Base-Rules

Answer 7

Let b be a positive real number with b

Question 7

Inverse relations for Exponential and Logarithmic Functions

Answer 7

bar any base b>0, with b≠1, the following inverse relations hold:
I1: b^logb^x=x, for x>0
I2: logb b^x=x, for real values of x