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Find the domain for each of the following functions: f ( x ) = ( 5 2 x ) / ( x 2 + 2 ) and g ( x ) = 5 x 1 .

The domain of f is (−∞, ∞). The domain of g is { x | x 1 / 5 } .

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Transcendental functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are sin x , cos x , tan x , cot x , sec x , and csc x . (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form f ( x ) = b x , where the base b > 0 , b 1 . A logarithmic function    is a function of the form f ( x ) = log b ( x ) for some constant b > 0 , b 1 , where log b ( x ) = y if and only if b y = x . (We also discuss exponential and logarithmic functions later in the chapter.)

Classifying algebraic and transcendental functions

Classify each of the following functions, a. through c., as algebraic or transcendental.

  1. f ( x ) = x 3 + 1 4 x + 2
  2. f ( x ) = 2 x 2
  3. f ( x ) = sin ( 2 x )
  1. Since this function involves basic algebraic operations only, it is an algebraic function.
  2. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
  3. As in part b., this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.
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Is f ( x ) = x / 2 an algebraic or a transcendental function?


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Piecewise-defined functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function    . The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of x :

f ( x ) = { x , x < 0 x , x 0 .

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for x < a and x > a , we need to pay special attention to what happens at x = a when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at x = a . We examine this in the next example.

Graphing a piecewise-defined function

Sketch a graph of the following piecewise-defined function:

f ( x ) = { x + 3 , x < 1 ( x 2 ) 2 , x 1 .

Graph the linear function y = x + 3 on the interval ( −∞ , 1 ) and graph the quadratic function y = ( x 2 ) 2 on the interval [ 1 , ) . Since the value of the function at x = 1 is given by the formula f ( x ) = ( x 2 ) 2 , we see that f ( 1 ) = 1 . To indicate this on the graph, we draw a closed circle at the point ( 1 , 1 ) . The value of the function is given by f ( x ) = x + 2 for all x < 1 , but not at x = 1 . To indicate this on the graph, we draw an open circle at ( 1 , 4 ) .

An image of a graph. The x axis runs from -7 to 5 and the y axis runs from -4 to 6. The graph is of a function that has two pieces. The first piece is an increasing line that ends at the open circle point (1, 4) and has the label “f(x) = x + 3, for x < 1”. The second piece is parabolic and begins at the closed circle point (1, 1). After the point (1, 1), the piece begins to decrease until the point (2, 0) then begins to increase. This piece has the label “f(x) = (x - 2) squared, for x >= 1”.The function has x intercepts at (-3, 0) and (2, 0) and a y intercept at (0, 3).
This piecewise-defined function is linear for x < 1 and quadratic for x 1 .
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Questions & Answers

discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
what is domain in calculus
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
oh sorry it's (-x^3+1)÷x^2
sorry I didn't understan A with that symbol
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
y= (2x^2+5)(3x+9)^2
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
eyad Amin that's the correct answer?
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eyad please answer my other question for my assignment
y= (2x^2+5)(3x+9)^2
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it's just the opposite of differential calculus
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Hello can someone help me with calculus one...
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
d the derivative of y= e raised to power x
rates of change and tangents to curves
Kyaw Reply
how can find differential Calculus
derivative of ^5√1+x
Rohit Reply
can you help with this f(×)=square roots 3-4
using first principle, find the derivative of y=cosx^3
RUBY Reply
Approximate root 4 without a calculator
Tinkeu Reply
Approximate root 4.02 without using a calculator
unit of energy
< or = or >4.00000001

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