Chapter 6 Inverse Functions
6.1 Inverse Functions
Practice • Find the inverse function of f (x) = x3 + 2. • Solution: • first write: • y = x3 + 2 • then solve this equation for x: • x3 = y – 2 • x =
cont’d • Finally, we interchange x and y: • y = • Therefore the inverse function is f–1(x) =
Practice: • If f (x) = 2x + cos x, find (f–1) (1). • Solution: • Notice that f is one-to-one because • f(x) = 2 – sin x > 0 • and so f is increasing. To use the theorem we need to know f–1(1): • F(0) = 1 f–1(1) = 0 • Therefore:
Exponential Functions • 6.2
Exponential Functions • The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function g(x) = x2, in which the variable is the base. • In general, an exponential function is a function of the form • f(x) = ax • where a is a positive constant.
Exponential Functions • The graphs of members of the family of functions y = axare shown in Figure 3 for various values of the base a. • Member of the family of exponential functions • Figure 3
Properties of f(x)=bx: • Domain: (- ∞, ∞) • Range: (0, ∞) • b0 = 1 • b > 1 f increasing • 0 < b < 1 f decreasing
Exponential Functions • Notice that all of these graphs pass through the same point (0, 1) because b0 = 1 for b ≠ 0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x > 0).
Natural exponential • The exponential function f(x) = ex is one of the most frequently occurring functions in calculus and its applications, so it is important to be familiar with its graph and properties. • The natural exponential function
Derivatives of Natural Exponential • In general if we combine Formula 8 with the Chain Rule, as in Example 2, we get
Example • Differentiate the function y = etan x. • Solution: • To use the Chain Rule, we let u = tan x. Then we have y = eu, so
Limits at infinity of exponential • We summarize these properties as follows, using the fact that this function is just a special case of the exponential functions considered in Theorem 2 but with base a = e > 1.
Example • Find • Solution: • We divide numerator and denominator by e2x: • = 1
Example – Solution • cont’d • We have used the fact that as and so • = 0
Integral of Natural Exponential • Because the exponential function y = ex has a simple derivative, its integral is also simple:
Example • Evaluate • Solution: • We substitute u = x3. Then du = 3x2 dx , so x2 dx = du and
Applications of • Exponential Functions
Applications of Exponential Functions • Table 1 shows data for the population of the world in the 20th century, where t = 0 corresponds to 1900. Figure 8 shows the corresponding scatter plot. • Scatter plot for world population growth • Table 1 • Figure 8
Applications of Exponential Functions • The pattern of the data points in Figure 8 suggests exponential growth, so we use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model • P = (1436.53) (1.01395)t
Applications of Exponential Functions • Figure 9 shows the graph of this exponential function together with the original data points. • Exponential model for population growth • Figure 9
Applications of Exponential Functions • We see that the exponential curve fits the data reasonably well. • The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s.
6.3 • Logarithmic Functions
Logarithmic Functions • If a > 0 and a 1, the exponential function f(x) = axis either increasing or decreasing and so it is one-to-one. It therefore has an inverse function f–1, which is called the logarithmic function with base a and is denoted by loga. If we use the formulation of an inverse function, • f–1(x) = y f(y) = x • then we have • Thus, if x > 0, then logax is the exponent to which the base a must be raised to give x.
Example • Evaluate (a) log3 81, (b) log25 5, and (c) log10 0.001. • Solution: • (a) log3 81 = 4 because 34 = 81 • (b) log25 5 = because 251/2 = 5 • (c) log10 0.001 = –3 because 10–3 = 0.001
Properties of f(x)=logbx: • Domain: (0, ∞) • Range: (- ∞, ∞) • logb1 = 0 logbb = 1 • b > 1 f increasing • 0 < b < 1 this definition for log is not used much and can be brought back to a base >1 (log1/bx = -logbx)
Logarithmic Functions • In particular, the y-axis is a vertical asymptote of the curve y = logax.
Example • Find log10(tan2x). • Solution: • As x 0, we know that t = tan2x tan20=0 and the values of t are positive. So by with a = 10 >1, we have
Natural Logarithms • The logarithm with base is called the natural logarithm and has a special notation: • If we put a = e and replace loge with “ln” in and , then the defining properties of the natural logarithm function become
Natural Logarithms • In particular, if we set x = 1, we get
Example • Find x if ln x = 5. • Solution1: • From we see that • ln x = 5 means e5 = x • Therefore x = e5.
Example – Solution 2 • cont’d • Start with the equation • ln x = 5 • and apply the exponential function to both sides of the equation: • eln x= e5 • But the second cancellation equation in says that eln x= x. Therefore x = e5.
Example • Evaluate log8 5 correct to six decimal places. • Solution: • Formula 7 gives
Graph of the Natural Logarithm and Exponential • Graphs of the exponential function y = ex and its inverse function, the natural logarithm function: • The graph of y = ln x is the reflection of the graph of y = ex about the line y = x.
Derivatives of Logarithmic Functions • The corresponding integration formula is • Notice that this fills the gap in the rule for integrating power functions: • if n –1 • The missing case (n = –1) is now supplied.
Example • Find ln(sin x). • Solution: