Unformatted text preview: Transcendental Functions Any function which is not algebraic is transcendental. The trig, exponential, and their inverses are examples of transcendental functions. A transcendental function is usually characterized by an inﬁnite series expansion, and there are a huge number of transcendental functions. Section 1.3: New Functions From Old Functions
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Vertical and Horizontal Shifts: Suppose c > 0, to obtain the graph of y = f (x) + c, shift the graph of y = f (x) a distance c units upward y = f (x) − c, shift the graph of y = f (x) a distance c units downward y = f (x − c), shift the graph of y = f (x) a distance c units to the right y = f (x + c), shift the graph of y = f (x) a distance c units to the left Vertical and Horizontal Stretching and Reﬂecting: Stretching: If c > 1 then the graph of y = cf (x) is the graph of y = f (x) stretched by a factor of c in the vertical direction. Reﬂection: The graph of y = −f (x) is the graph of y = f (x) reﬂected about the xaxis, because our ordered pair has changed from (x, f (x)) to (x, −f (x)). Chapter 1 Review 4 Suppose c > 1, to obtain the graph of
Chapter 1 Review y = cf (x) stretch the graph of y = f (x) vertically by a factor of c Suppose c > 1, to obtain the graph of 4 y = (1/c)f (x) compress the graph of y = f (x) vertically by a factor of c y = f (cx) compress the graph of y = f (x) horizontally by a factor of c y = f (x/c) stretch the graph of y = f (x) horizontally by a factor of c y = cf (x) stretch the graph of y = f (x) vertically by a factor of c y = (1/c)f (x) compress the graph of y = f (x) vertically by a factor of c y y = (cx)(x) reﬂect the graph of y y = (x(xhorizontally byaxis = f −f compress the graph of = f f ) ) about the x a factor of c y y = (x/c) stretch the graph of of = f (x)(horizontally by a factor of c = f f (−x) reﬂect the graph y y = f x) about the y axis Algebraic Combinations of Functions y = −f (x) reﬂect the graph of y = f (x) about the xaxis y = f (−x) reﬂect the graph of y = f (x) about the y axis Algebraic Combinations of Functions
(f + g )(x) = f (x) + g (x) domain A ∩ B (f − g )(x) = f (x) − g (x) domain A ∩ B (f g )(x) = f (x)g (x) domain A ∩ B
f g Let f and g be functions with domains A and B . Then the functions f + g, f − g, f g, f /g are deﬁned as: Let f and g be functions with domains A and B . Then the functions f + g, f − g, f g, f /g are deﬁned as: (f + g )(x) = f (x) + g (x) domain A ∩ B (f − )(x) = f (x) − g (x) domain A ∩ B g (f g )(x) = f (x)g (x) domain A ∩ B
f (x) = f (x) g (x) domain = {x ∈ A ∩ B g (x) = 0} g Graphical g(x) Addition We can add functions graphically by by simply adding their corresponding y coordinates. (x) = f (x) domain = {x ∈ A ∩ B g (x) = 0} Graphical Addition We can add functions graphically by by simply adding their corresponding y coordinates. Composition of functions Composition of functions composite function f ◦ g (called the composition of f and g ) is deﬁned by Given two functions f and g , the
Given two functions f and g , the composite function f ◦ g (called the composition of f and g ) is deﬁned by (f ◦ g )(x) = f (g (x)) It is important to note that f ◦ g (x) = g f ◦ gx)!! = f (g (x)) ( ◦ f ()(x) It is important to note that f ◦ g (x) = g ◦ f (x)!! Graphical Composition To ﬁnd the composition f ◦ g (a) from a graph or table, we can use the following method (if the graphgiven, g (a), and then use the value and )apply a best ﬁt line). From a table is read create graphs of the functions g (a as the input to determine f (g (a)). You can build From the graph read g (a), and then use the value g (a) as the input to determine f (g (a)). You can build a table of values this way, and then plot f (g (x)). Graphical Composition To ﬁnd the composition f ◦ g (a) from a graph or table, we can use the following method (if a table is given, create graphs of the functions and apply a best ﬁt line). a table of values this way, and then plot f (g (x)). Chapter 1 Review 5 Example of Multiple Compositions (f ◦ g ◦ h)(x) f (x) = x , g (x) = x10 , h(x) = x + 3 x+1 = f (g (h(x))) = f (g (x + 3)) = f ((x + 3)10 ) (x + 3)10 = (x + 3)10 + 1 (f ◦ g ◦ h)(x) Example of Decomposing using composition Given L(x) = cos2 (x + 9), determine functions f, g, h so you can write L(x) as a composition (f ◦ g ◦ h)(x) (that is, L(x) = (f ◦ g ◦ h)(x)). SOL: Look at how you compute L(x), and build the functions from that: 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 ...
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 Fall '08
 JUNGHENN
 Calculus, Algebra

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