# Graph question

## Homework Statement

is it possible to have a tangent line in a cubed function

## The Attempt at a Solution

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Simply put, if you can differentiate it, it has tangent lines. So you can have tangent lines for things that aren't functions too.

but the tangent line touches a cubed function twice so im sure if it could really be called a tangent line

Mark44
Mentor
but the tangent line touches a cubed function twice so im sure if it could really be called a tangent line
Doesn't matter. The tangent line is just a line that touches a curve at a point (a, f(a)) and whose slope is f'(a). The fact that the tangent line happens to intersect the graph of the function somewhere else is immaterial. Pretty much every odd-degree polynomial will have a tangent line that intersectst the curve somewhere else.

As it turns out, the tangent line to the graph of y = f(x) = 2x + 3 at any point happens to completely coincide with the graph of this function, but that doesn't keep it from being a tangent line.

jbunniii
In addition to what Mark44 said, I will point out that the tangent line can even intersect/cross the curve AT the point of tangency. For example, the tangent to $f(x) = x^3$ at $x = 0$. It's still a tangent line, though.