Question

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

Answer 1

`f(x) = x^-4` can also be written in the form `f(x) = 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(x)` 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 `(0, oo)`, then it goes down sharply until it reaches `(1, 1)`, and finally it decreases sharply approaching `(oo, 0)` .

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`