If I Is Raised to an Odd Power, Then It Can Not Simplify to Be

Power Functions

Algebraic Representation

It is easy to confuse power functions with exponential functions. Both have a bones form that is given past two parameters. Both forms look very like. In exponential functions, a fixed base is raised to a variable exponent. In power functions, nevertheless, a variable base of operations is raised to a fixed exponent.

The parameter a   serves equally a simple scaling cistron, moving the values of  x ^b up or down as  a  increases or decreases, respectively

The parameter b , called either the exponent or the power, determines the function'southward rates of growth or disuse. Depending on whether it is positive or negative, a whole number or a fraction,  b  will also determine the function'due south overall shape and beliefs.

More so than other simple families like lines, exponentials, and logs, members of the power family can exhibit many distinctive behaviors.

Observe that when  b = 0 the function simplifies to  f(x) = ax ⁰ = a1 = a , a constant function with an output of  a  for every input. When  b > 0 ,  f(0) = a0 ^b = 0. That is, every power office with a positive exponent passes through  (0, 0) . When  b < 0 ,  f(0)  is undefined. (Call up,  ten ^–b = 1 / x ^b .) Thus, power functions with negative exponents accept no y-intercepts. Power functions with negative, whole number exponents like  x ^–ane or  x ^–2 are simple examples of rational functions, and for these functions  10 = 0 is an instance of a singularity.

To appreciate the variety of behaviors among members of the power family, consider two elementary cases:

• Fifty-fifty powers.

  If  b  is a an even whole number like

   b = –2, four, 10, etc., and so for whatsoever input  x  nosotros volition accept  f(–x) = a(–ten) ^b = a(–ane) ^b (ten) ^b = a(x) ^b = f(x) , since  –1  raised to an fifty-fifty power is  1 . The part has a sure symmetry: Its outputs for any  ten  are exactly the aforementioned as its outputs for  –x . We call any function with this behavior an even function, with fifty-fifty powers serving as the archetype.
• Odd powers.

  If  b  is a an odd whole number like

   b = –1, 3, 7, etc., then for any input  10  we will have  f(–x) = a(–10) ^b = a(–i) ^b (10) ^b = a(–1)(x) ^b = –f(ten) , since  –1  raised to an odd power is  –ane . The function has a certain anti-symmetry: Its outputs for any  x  are exactly the reverse of its outputs for  –x  . Nosotros call any function with this behavior an odd function, with odd powers serving as the archetype.

The difference between odd and even powers just hints at the differences among power functions.

Another helpful stardom separates functions with whole number (integer) powers from those with fractional powers. (Nosotros get out the consideration of irrational powers to calculus.)

• Integer powers.

  We've already noted the symmetry/anti-symmetry of even/odd integer powers. There is also a primal difference between positive and negative integer powers. We've noted that all positive powers pass through

   (0, 0), while all negative powers have a singularity at  x = 0 . When we put together the even/odd possibilities with the positive/negative possibilities for integer powers, we find four distinct cases for growth and decay.

  Cases for integer powers:

• Fractional powers.

  It doesn't make any sense to distinguish between "even" and "odd" fractional powers – those terms refer but to integers.

  It does makes sense to talk about positive and negative fractional powers, even so, and this distinction is over again important in determining overall behavior.

  Another distinction – a new one – likewise comes into play when we consider fractional powers. Suppose that the fractional power  m/n  has been reduced to lowest terms (all mutual factors in the numerator and the denominator take been cancelled). To calculate  x ^m/northward , we proceed in two steps: one) Notice the  n ^th   root of  x (x ^1/n = ); 2) Enhance information technology to the  m ^th   power. The 2nd step is straightforward: Nosotros can heighten any number to an integer power. The outset step, withal, is problematic for negative  x : We can not meaningfully ask for an even root of a negative number. Thus, the commanded inputs (domains) of partial powers depend on whether  n  is even or odd.

  Considering this in combination with the positive/negative possibilities, we once over again find 4 distinct cases for growth and disuse (each with several sub-cases).

  Cases for partial powers:

It is difficult to unmarried out one key  algebraic holding of ability functions from all of this diverseness. Mayhap it is best to say simply that powers behave like powers, with an appreciation for the many unlike behaviors that this can encompass.

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Source: http://wmueller.com/precalculus/families/1_41.html

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