1.6 - Inverse Functions and Logarithms

c Kathryn Bollinger, September 11, 2010 1

Inverse Functions

Def: A function f is said to be a one-to-one function if it never takes on the same value twice; that is f(x1) = f(x2) whenever x1 = x2 6 6 (Each range value corresponds exactly to one domain value.)

Ex: Which of the following represents a one-to-one function?

Horizontal Line Test: A function is a one-to-one function if and only if no horizontal line inter- sects its graph more than once.

Ex: Are the following one-to-one functions?

f(x) g(x) 5 4

−4 −2 2 4 −5 5

−4 −5

Ex: Are exponential functions of the form f(x)= ax (where a> 0 and a = 1) one-to-one? 6

Note: Any continuous function that is either increasing or decreasing for all domain values is a one-to-one function. c Kathryn Bollinger, September 11, 2010 2

Def: If f is a one-to-one function with domain A and range B, then its inverse function f −1 has domain B and range A and is deﬁned by

− f 1(y)= x f(x)= y ⇐⇒ for any y in B. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of f −1.

NOTE: If f is one-to-one, then the graph of its inverse, f −1, is a reﬂection about the line y = x. If f is not one-to-one, then f has no inverse.

Ex: If f is a one-to-one function and f(3) = 4 and f(10) = 8, ﬁnd − (a) f −1(4)

(b) f(f −1( 8)) −

Finding the Inverse of a One-to-One Function

1. Write y = f(x).

2. Solve this equation for x in terms of y (if possible).

3. To express f −1 as a function of x, interchange x and y. The resulting equation is y = f −1(x).

Ex: Find the formula for the inverse of f(x)= √2x + 4. What are the domain and range of f and f −1? c Kathryn Bollinger, September 11, 2010 3

Logarithmic Functions

If b > 0 and b = 1, the exponential function f(x) = bx is a one-to-one function, and so it has an 6 inverse function which is called a logarithmic function. Def: Let b be a positive number with b =1. If x > 0, the logarithmic function with base b, 6 denoted by logb x, is deﬁned as y = log x by = x b ⇐⇒

** logb x is the exponent to which b must be raised in order to obtain x.**

Ex: Graph f(x) = 10x and its inverse on the same graph.

1 x Ex: Graph f(x)= and its inverse on the same graph. 2 c Kathryn Bollinger, September 11, 2010 4

Common Logarithm LOG : b =10= y = log x = log x 10y = x ⇒ 10 ⇐⇒

Natural Logarithm LN : b = e = y = log x = ln x ey = x ⇒ e ⇐⇒

log a ln a Change of Base Formula: log a = = b log b ln b

Ex: Change each exponential equation to an equivalent logarithmic form:

(a) 32 = 9

(b) 10−3 = 0.001

Ex: Solve for x,y, or b without a calculator:

(a) log x = 3 −

(b) ln x = 0

(d) log3 81 = y

Ex: Find the value of log6 5, correct to six decimal places.

Ex: Find the domain of f(x) = 5 + log (2x + 3). c Kathryn Bollinger, September 11, 2010 5

Properties of Logarithmic Functions: Let b> 0, b = 1, where m and n are positive real numbers and r is any real number. Then, 6 log (mn) = log m + log n • b b b m log = log m log n • b n b − b log mr = r log m • b · b log 1 = 0 • b log b = 1 • b x = blogb x if x> 0 • x = log bx • b log m = log n if and only if m = n • b b

Ex: Write 1 (ln 2 + ln x ln y) as a single logarithm. 2 −

Ex: Given logb 2 = 1.346 and logb 5 = 3.876, ﬁnd the following:

(a) logb 50

4 (b) log b b2 c Kathryn Bollinger, September 11, 2010 6

Ex: Solve the following for x EXACTLY:

(a) 3x = 7

(b) 2 103x = 5 ·

(d) log3 x + log3 (x + 2) = 1

(e) log(ln 4x) = 0 c Kathryn Bollinger, September 11, 2010 7

(f) log(3 x) log(2x + 9) = 2 − −

Ex: Sketch the graph of the function y = ln(x 3) + 1 using transformations of a basic function. − c Kathryn Bollinger, September 11, 2010 8

Applications

Logarithmic Regression •

In order to ﬁnd the best ﬁtting logarithmic model to a set of data, use the option: 9:LnReg for a logarithmic model of the form y = a + b ln x under STAT on your calculator.

Interpreting an Inverse Function •

Ex: Recall from Section 1.5 that we showed the mass of Iodine-131 that remains from a 100 t/8 gram sample after t days is m(t) = 100 1 grams. Find the inverse of this function and · 2 interpret it.