Question

Two functions are graphed below. One function is
f
(
x
)
=
x
n
, and the other function is
g
(
x
)
=
n
x
for some whole number, n, greater than 1. Point A is at
(
n
,
n
)
.



image.png

Which conclusion must be true?


The dashed curve represents
f
(
x
)
because
f
(
x
)
>
g
(
x
)
for
x
<
n
.


The solid curve represents
g
(
x
)
because
f
(
x
)
<
g
(
x
)
for
x
>
n
.


The solid curve represents
f
(
x
)
because
f
(
x
)
≤
g
(
x
)
over the interval
|
0
,
n
|
.


The dashed curve represents
g
(
x
)
because
g
(
x
)
rises slower than
f
(
x
)
over the interval
[
n
,
∞
]
.

Answer 1

I need one clarification (the image you referenced is not visible to me): is g(x) = n^x (exponential) or g(x) = n·x (linear)? The typed line "g(x) = n x" could mean either.

As written (with g(x) = n·x) the point A = (n,n) is inconsistent with f(x) = x^n (since f(n) = n^n ≠ n), so the problem must mean g(x) = n^x and A is really at (n, n^n) (the superscript may have been lost).

Even assuming the intended pair is f(x) = x^n and g(x) = n^x, none of the four given conclusions is true for every whole number n > 1. Quick counterexamples:

- Option 1 ("f(x) > g(x) for x < n"): false for n = 2 (e.g. at x = 1.5, 2^1.5 > 1.5^2).
- Option 2 ("f(x) < g(x) for x > n"): false for n = 2 (e.g. at x = 3, 3^2 > 2^3).
- Option 3 ("f(x) ≤ g(x) over [0,n]"): false for n = 10 (e.g. at x = 9, 9^10 > 10^9).
- Option 4 ("g rises slower than f on [n,∞]"): false in general (for n ≥ 3 the exponential grows faster immediately and certainly for large x).

So with the natural interpretation g(x) = n^x, there is no single choice that must be true for every integer n > 1. If you can confirm the exact functions and upload the graph, I will identify which statement matches the pictured curves.