Unformatted text preview: Add 9: h(x) = x + 9 Take cosine: g (x) = cos x Square: f (x) = x2 Section 1.5: Exponential Functions
Exponential functions are of the form f (x) = ax where a is a positive constant. Increasing, Decreasing or Constant Increasing, if 1 < a < ∞, Decreasing, if 0 < a < 1, Constant, if a = 1. The exponential function f (x) = ax will either be For the increasing exponential, the function will increase more rapidly (as x → ∞) the larger a gets. All exponential functions pass through the point (0, 1) (unless they are shifted, of course!). If a = 1 the domain is all reals and the range is (0, ∞). Laws of Exponents If a and b are positive numbers and x and y are any real numbers, then ax+y ax−y (ax )y (ab)x = = = = ax ay ax a−y axy ax bx Chapter 1 Review 6 Applications of Exponential Functions: Population Growth Suppose that we know a population doubles every hour. Say the population is p(t), and t is the time in hours. If our initial population is p0 , then we have: p(0) p(1) p(2) p(3) = = = = p0 2 p0 2p(1) = 22 p0 2p(2) = 23 p0 p(t) = 2t p0 And we see how the exponential function enters the ﬁeld of population growth. Q: What if the population tripled every hour, how would that change our ﬁnal answer? A: p(t) = 3t p0 . Applications of Exponential Functions: Half-Life The half-life is deﬁned for radioactive compounds to be the time it takes for the amount to decay into half that amount. The half-life for strontium-90 (90 Sr) is 25 years. If we have 24mg of left after t years. m(0) = 24 1 m(25) = (24) 2 1 m(50) = m(25) = 2 1 m(50) = m(75) = 2 m(t) =
90 Sr, ﬁnd an expression for amount 1 m(0) 22 1 m(0) 23 1 (24) = 24 · 2−t/25 2t/25 There was some talk about the number e More on that later with Logarithms and natural Logarithms. Section 1.6: Inverse Function and Logarithms
Not all functions have inverses. Recall that a function is deﬁned as a rule f that assigns to each element x in a set A exactly one element, called f (x) in a set B . This requirement that a function have exactly one element imposes a condition on whether or not a function f has an inverse function. Deﬁnition A function f is called a one-to-one function if it never takes on the same value twice, that is f (x1 ) = f (x2 ) whenever x1 = x2 . You can check whether or not a function is one-to-one from the graph by using the horizontal line test. ...
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