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MATH 161, Fall 2022, Day 2. Ch. 1.5 Inverse and Logarithmic Functions
Domain and Range
(See Ch. 1.1 for more detail.)
Definition: A function f is a rule that assigns to each real number in a set D exactly one real number, called f (x). (Functions will be defined more broadly in subsequent math classes.) The set D is called the domain of f. The range of f is the set of all possible values of f (x) as x varies throughout the domain.
Examples Let f (x) =
x โ 2.
- Find the domain of f. (โFind the domainโ means to find the largest subset D of the real numbers such that f (x) is also a real number for every x in D.)
- Find the range of f.
- Let g(x) =
x โ 2. This time define the domain D as the interval [2, 6]. We call g the restriction of f to D. Find the range of g.
- Find the range of
x โ 2
Ch. 1.5, Part 1: Inverse Functions
A function answers this question: Given a number a in the domain of f , what is the b satisfying f (a) = b? For many functions, there is an associated inverse function that answers this question: Given a b in the range of f , what is the a satisfying f (a) = b.?
If f has an inverse function, it is denoted f โ^1.
Some functions are not invertible! For example, suppose f (x) = x^2. What would f โ^1 (4) be? If f โ^1 is to be a function, it canโt have two possible outputs for a single input.
In particular, an invertible function has to be one-to-one.
Definition: A function f is called one-to-one if it never takes the same value twice. That is
f (a) = f (b) only if a = b.
Examples: g(t) = t^2 and f (x) = 6 are not one-to-one. Show this by citing specific instances where f (a) = f (b), but a 6 = b.
The graph of a one-to-one function passes the horizontal line test.
If a function is one-to-one, we may invert it, because if an input x gives exactly one output y, then we can reverse the process and still get a well-defined function.
Definition: Let f be a one-to-one function with domain A and range B. Then its inverse function, denoted f โ^1 , has domain B and range A and is defined by f โ^1 (y) = x if and only if f (x) = y.
Caution! f โ^1 (x) 6 =
f (x)
in general.
In general, to invert a function and find an explicit f โ^1 (x):
- Write y = f (x).
- Solve this equation for x as a function of y.
- To express f โ^1 as a function of x, interchange the x and the y. Then weโll get y = f โ^1 (x).
Example: Let f (x) = 3x + 2. Show that f is one-to-one. Find f โ^1.
Example: Find f โ^1 (x) if
f (x) =
x + 1 x + 2
Find the domain and range of f and of f โ^1.
To obtain the graph of an inverse function, reflect the graph of the function across the line y = x. That is, if (a, b) is on the graph of f , then f (a) = b. So f โ^1 (b) = a. Then (b, a) is on the graph of f โ^1.
Example: Graph f (x) = 3x + 2 and its inverse on the same pair of axes. Draw the line y = x. Where do the three lines intersect?
Ch. 1.5, Part 2: Logarithms
The inverse of an exponential function is called a logarithm.
Definition: The function logb is the function satisfying
logb x = y if and only if by^ = x
The range of f (x) = bx^ is (0, โ), and its domain is (โโ, โ). So the domain of logb is (0, โ) and its range is (โโ, โ).
Examples:
- Find log 4 (1)
- Find log 6 (1)
- Find log 3 (9)
- Find log 5 (25)
- Find log 4 (2)
- Find log 6 (1/6)
Example: Evaluate log 2 80 โ log 2 5.
The Natural Logarithm We call loge the natural log. We write ln instead of loge.
ln x = y if and only if ey^ = x
As always, ln(ex) = x for any x and eln^ x^ = x for positive x. Also, ln e = 1, because e^1 = e, and ln 1 = 0, because e^0 = 1.
Example: Solve: e^5 โ^3 x^ = 10
