# What is the graph of f(x)=x^-4?

Oct 11, 2014

$f \left(x\right) = {x}^{-} 4$ can also be written in the form $f \left(x\right) = \frac{1}{x} ^ 4$

Now, try substituting a some values

f(1) = 1
f(2) = 1/16
f(3) = 1/81
f(4) = 1/256
...
f(100) = 1/100000000

Notice that as $x$ goes higher, $f \left(x\right)$ goes smaller and smaller (but never reaching 0)

Now, try substituting values between 0 and 1

f(0.75) = 3.16...
f(0.5) = 16
f(0.4) = 39.0625
f(0.1) = 10000
f(0.01) = 100000000

Notice that as $x$ goes smaller and smaller, f(x) goes higher and higher

For $x > 0$, the graph starts from $\left(0 , \infty\right)$, then it goes down sharply until it reaches $\left(1 , 1\right)$, and finally it decreases sharply approaching $\left(\infty , 0\right)$ .

Now try substituting negative values

f(-1) = 1
f(-2) = 1/16
f(-3) = 1/81
f(-4) = 1/256

f(-0.75) = 3.16...
f(-0.5) = 16
f(-0.4) = 39.0625
f(-0.1) = 10000
f(-0.01) = 100000000

Since the exponent of $x$ is even, the negative value is removed.

Hence, for $x < 0$, the graph is a mirror image of the graph for $x > 0$