Recognizing the common types of functions

Whenever we can classify a function as one of the familiar types such as quadratic or trigonometric, we know it will have certain well-known properties. For example, a trigonometric function will be periodic. As you learn to use the vocabulary of functions more and more precisely, it will be easier for you to read mathematics with genuine understanding. When you are first presented with any formula, you should try to establish what type of a function it describes. This may require you to rewrite the function in order to convert it into a standard form.

Linear functions

x, (1/2)t, -0.4P, R+3, 7s+4.6, 4(z-1)+2, pi-2theta, etc.
The form of a linear function is easily recognizable:
f(x) = ax + b, where a, b are constant

Quadratic functions

x^2, .0001L^2, -(1/2)r^2, x^2 - 4, x(x+2), 3x^2+x+1, etc.
A function is quadratic if it can be put in the form
f(x) = ax^2 + bx + c, where a, b, c are constant
Note also that a must be non-zero -- if a is zero, we just have a line.

Cubic functions

x^3, 1+5t^3, x^3+2x^2-x+(1/7), 3q(q^2+2), (P+2)^3, etc.
A function is cubic if it can be put in the form
f(x) = ax^3 + bx^2 + cx + d, where a, b, c, d are constant
Note again that a must be non-zero.

Polynomial functions

4x^5+3x^4+2x^2+1, 1+M+M^4, (1/2)l(l-1)^3, -x^2+16.2x, etc.
A polynomial function can be written in the form
f(x) = an x^n + a{n-1} x^(n-1)+...+a1 x + a0,
where n is a non-negative integer (note that this means that all of the exponents are integers), and all of the coefficients are constants. Of course, linear, quadratic and cubic functions are all also polynomials.

Rational functions

1/(1+x), (x^2+x)/(3x^3+2x+14), (1-t)/(t^2-3t+4), p^3/(2p+7), etc.
All of the four preceding types (linear, quadratic, cubic, polynomial) are special cases of the broader category called rational functions, which is made up of all quotients of polynomials. The rational functions can be written in the form
f(x) = p(x)/q(x), where p, q are polynomials

Power functions

x^2, x^3, t^(-2), z^(1/2), 2R^9, sqrt(s), .03x^(2/3), T^4, x^pi, etc.
A power function is a function formed by raising a variable to a constant power and then multiplying by a constant -- which may be one. Power functions can therefore be written in the form
f(x) = ax^r, where a, r are constant
Note that a, and r are real numbers.

Exponential functions

10^x, e^x, 2^(-x), 100 e^(2t), 0.001(1.07)^t, etc.
These functions are formed in a different way from power functions. An exponential function is a constant raised to a variable power (and then multiplying by a constant). An exponential function can therefore be written in the form
f(x) = a (b)^x, where a, b are constant
Note that b must be positive.

Logarithmic functions

log(x), ln(x), 56 log(3x), (1/2) ln(x), etc.
These are easy to recognize because the name of the function is always included in the expression. The definitions are

f(x) = log(x)   the inverse of the exponential 10^x
g(x) = ln(x)   the inverse of the exponential e^x

Remember that an expression like ln(3) is a constant!

Trigonometric functions

sin(x), cos(x), tan(x), sin(pi t), 4cos(2theta - pi/2), etc.
Trigonometric functions are designated by name in mathematical writing. The definitions, which refer to the unit circle, are
figure showing circle with sine, cosine and tangent shown

Examples
  1. Which of the following are polynomials?
    f(x) = 3x^3+2x^2+x^(1/2)+2
    g(x) = 5x^2 - x^(-1)
    Polynomials do not have terms with negative exponents or fractional exponents, so neither f(x) nor g(x) is a polynomial.
  2. Which of the following are rational functions?
    f(t)=(t^2+2)/sqrt(1+t)
    g(y)=3y(y+1)^(-2)
    The function f(t) is not a rational function because the denominator is not a polynomial. On the other hand, the function g(y) can be rewritten as a fraction with numerator 3y and denominator (y+1)(y+1), both of which are polynomials, so it is a rational function. (Is the denominator obvious? Expand it to see!)
  3. What type of function is h(x)?
    h(x) = (e^x)^5
    We must decide whether h(x) has the exponential form of a constant to a variable power or the power form of a variable to a constant power. Using the rules of exponents, we can write
    h(x) = (e^x)^5 = e^(5x) = (e^5)^x
    Thus h(x) is a constant, e^5, raised to a variable power. So it is an exponential function.
  4. What type of function is l(q)?
    l(q) = 2q(ln e^q + 1)
    The logarithm term simplifies to just q, so we have
    l(q) = 2q(ln e^q + 1) = 2q(q+1)
    which is a quadratic function.
Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

For each, select all of the function types that apply.

  1. 3x^2+1   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  2. 3^(x-1)+4   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  3. 4t^3+(1/pi)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  4. 3y^(-2)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  5. 4x + 5e   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  6. 12(1-0.04)^(t-1)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  7. (2z)^(1-0.04)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  8. (5x-x^(-1/2))/(x+4)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
  9. (x+2)^(-2)/(x-1)^(-3)   is  
    linear quadratic cubic polynomial
    rational power exponential logarithm

    (Enter answers, then click: . Answer message: )
next page
section test

precal: 13.1 - recognizing functions
page created: Fri Nov 28 05:44:47 2014
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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan