In this question we're asked which one
of the following functions is not a power function?
So, let's begin by reviewing the definition
of a power function.
A power function is a function that can be
represented in the form f of x equals k
times x raised to the power of p,
where the exponent p is any real number
and k, the coefficient, cannot equal zero.
Notice if k was zero, we would just have
the function f of x equals zero,
which would be a constant function.
So looking at our first function,
we have y equals one over x, or one divided by x.
This does not look like a power
function in its current form.
But, we can rewrite this using the exponent
property given here, where a raised to
the power of negative m equals one over a
raised to the power of positive m.
To help remember this exponent property,
we can write a raised to the power of negative m
as a fraction with a denominator of one.
And if we move this exponential expression down to
the denominator, it changes the sign of the exponent.
Again, a raised to the power of negative m
equals one over a raised to the power of positive m.
Similarly, if we have one over a raised to the power
of negative m, this is equal to a raised to
the power of positive m, which again we can think
of as a fraction with a denominator of one.
So if we move this up to the numerator,
it changes the sign of the exponent.
So if we have an exponential expression
and we move it across the fraction bar,
it will change the sign of the exponent.
So going back to our first function,
we have y equals one over x,
which is the same as one over x
raised to the power of positive one.
So, they moved this up to the numerator.
We can rewrite this function as y equals
x raised to the power of negative one.
So notice in this form, p is equal to
negative one, the exponent, and k, the coefficient,
would be equal to positive one.
And therefore, this is a power function.
Next, we have y equals 3x squared.
So, this does fit the form of a power function,
where p equals two and k equals three.
Now the next function, we have y equals x squared plus one.
Because of this plus one here on the end,
this function is not a power function.
The form of a power function does not allow
the addition or subtraction of additional terms,
or in this case, plus or minus a constant.
So, this is our function that's not a power function.
Let's also take a look at our last function,
where we have y equals the square root of x.
Again in its current form,
this does not appear to be a power function.
But, we can rewrite a radical using
rational exponents using the property given here,
where the nth root of a raised to the power of m
is equal to a raised to the power of m over n,
over m divided by n.
So, the index gives us the denominator
and the exponent gives us the numerator.
So for the square root of x, the index is two
and the exponent, x, is equal to one.
And so, we can rewrite this as y equals x
raised to the power of 1/2.
So in this form, notice how the exponent, p,
is equal to 1/2 and the coefficient, k,
is equal to one, and therefore,
this is a power function.
So, the only function here that's not
a power function is y equals x squared plus one.
Now before we go, let's look at several
other power functions and their graphs.
Here, we have f of x equals x,
f of x equals x squared,
f of x equals x cubed,
f of x equals one over x,
which equals x raised to the power of negative one,
f of x equals one divided by x squared,
which equals x raised to the power of negative two,
as well as f of x equals the square root of x,
which equals x raised to the power of 1/2.
So, notice how our power functions
give us many of our basic functions
for what we often call parent functions.
I hope you found this helpful.