5.4 Transformations and Composition of Functions

Transcription

1 5.4 Transformations and Composition of Functions 1. Vertical Shifts: Suppose we are given y = f(x) and c > 0. (a) To graph y = f(x)+c, shift the graph of y = f(x) up by c. (b) To graph y = f(x) c, shift the graph of y = f(x) down by c. g(x) = x 2 +3 h(x) = x Horizontal Shifts Suppose we are given y = f(x) and c > 0. (a) To graph y = f(x c), shift the graph of y = f(x) to the right by c. (b) To graph y = f(x+c), shift the graph of y = f(x) to the left by c. g(x) = (x 6) 2 h(x) = x+4 3. Reflections To graph y = f(x), reflect the graph of y = f(x) across the x-axis. g(x) = x h(x) = e x 1

2 4. Vertical Stretching and Shrinking To graph y = cf(x): (a) If c > 1, stretch the graph of y = f(x) vertically by a factor of c. (b) If 0 < c < 1, shrink the graph of y = f(x) vertically by a factor of c. g(x) = 3x 2 h(x) = 1 2 x Give the parent function and list the transformations needed to graph the functions below. g(x) = 3(x+2) 2 f(x) = x +3 h(x) = e x 4 5 If the graph of f(x) = x is shifted left 3 units, shrunk vertically by a factor of 2 7, reflected across the x-axis, and then shifted up 2 units, what is the equation of the resulting graph? We can also combine functions by finding the composition of two functions. If f and g are two functions, the composition of f and g is the function h(x) = f(g(x)) In other words, we evaluate g first, and then we plug in whatever we get into f. For f(x) = x+2 and g(x) = x 2, find f(g(2)) and g(f(2)). 2

3 Consider the graphs of f and g below. Find the indicated compositions. f(g(5)) g(f(3)) f(g( 1)) g(f(0)) 5.5 Logarithmic Functions To solve an equation like x 3 = 8, we take the cube root of both sides and get that x = 2. So we would say that the function f(x) = x 3 and the cube root function g(x) = 3 x are inverse functions because cubing something and cube-rooting undo each other. Problem: How do we solve an equation like 3 x = 7? What can we do to both sides that undoes the exponential 3 x? The answer is to use a new type of function called a logarithmic function. The logarithmic function with base a is denoted f(x) = log a x and it has the following key relationship with exponentials: log a x = y is equivalent to a y = x log a x is the exponent to which a must be raised to get x. log 2 8 = 3 since 2 3 = 8 log = 2 since 3 2 = 1 9 Evaluate the following expressions. log 4 16 log log 9 3 log 3 1 log 5 0 log 5 ( 25) 3

4 The logarithmic function with base 10 is called the common logarithm and usually instead of writing log 10 x, we just write logx. log1000 log 1 10 The logarithmic function that has base e is called the natural logarithmic function. Instead of writing log e x, we write lnx. lne ln1 lne 6 To evaluate logarithms on your calculator, you can use the following Change of Base formula if your calculator does not have the ability to do logs with other bases. log a x = lnx lna Evaluate log 5 33 to 4 decimal places. In general, though, you should leave your answer in terms of logarithms if they do not evaluate to an exact number. Only round if the problem tells you to! Cancellation Rules: log a a x = x a log a x = x lne x = x e lnx = x The graph of f(x) = lnx is below. The graphs of logarithmic functions with base a > 1 all have the same general shape. As we saw above, you CANNOT take the logarithm of a negative number or 0. You can ONLY take the log of a POSITIVE number! The domain of f(x) = log a x is (0, ) (for any base a). 4

5 To find the domain of more complicated logarithmic functions, set whatever is inside the logarithm greater than 0. Find the domain of the function f(x) = ln(7x+6). Find the domain of the function f(x) = log 5 (8 2x). Properties of Logarithms: 1. log a (xy) = log a x+log a y ( ) x 2. log a = log y a x log a y 3. log a (x c ) = clog a x THE ONLY WAY TO COMBINE/EXPAND LOGARITHMS IS BY USING ONE OF THESE THREE PROPERTIES. IF YOU ARE NOT USING ONE OF THESE PROPERTIES, DON T DO IT!! Expand the following as much as possible using properties of logarithms. ( ) x 2 log a yz 3 ( ln 2x 5 ) y ( ) x 3 ln (x+1) 10 5

6 Suppose you are given the following: log a 2 = 0.73 log a 5 = 1.68 log a 7 = 2.04 Use the above to evaluate the following: log a 10 ( ) 35 log a 4 log a (4a 6 ) 7 Use properties of logarithms to combine the following expressions into a single logarithm. 3logx+alogy alnx+2ln4 1 3 lny Solving Logarithmic Equations The general strategy to solving logarithmic equations is to rewrite the equation in exponential form using the fundamental relationship log a x = y a y = x Often, you first need to combine all logarithms on one side of the equation. Examples log 3 (x 2 5x 5) = 2 6

7 ln(2 5x) 4 = 0 log(2x+12) 1 = log(2 x) Solving Exponential Euqations The general strategy to solving exponential equations is to isolate the exponential expression on one side of the equation and then take the natural logarithm of both sides. Simplify using logarithm properties and solve for the variable. Examples: 2e 12x+4 = x = x/7 9 = 0 Solve by factoring: x 2 e x +5xe x +6e x = 0 7

8 Applications: If $6500 is invested in an account at 8% per year compounded continuously, how long will it take for the amount in the account to be $7500, rounded to 3 decimal places? Suppose the number of a certain species of fish is modeled by the function n(t) = t where t is measured in years. What is the initial number of fish? When will the number of fish reach 30, rounded to 3 decimal places? 8