Can You Find a Function from Its Tangent Line?

A tangent line tells you how a function behaves at one point: it gives the function’s value there and its instantaneous slope. It generally does not tell you the function’s values everywhere else.

Suppose a line is tangent to the graph of $f$ at $x=a$. You can use the line to write two conditions on $f$. Those conditions are useful, but many different functions can satisfy them. To find one particular function, you need more information or an assumption about the form of the function.

What a tangent line tells you

Write the tangent line as $y=mx+b$, where $m$ is its slope. If it touches the graph at $x=a$, then the function must have the same height and slope as the line at that point:

$$f(a)=ma+b, \qquad f'(a)=m.$$

The first condition says the graph passes through the point $(a,ma+b)$. The second says its derivative at $a$ equals the line’s slope. These are the defining conditions for the line to be tangent there, assuming $f$ is differentiable at $a$.

Why the function is usually not unique

Those conditions constrain the function at just one input. You can change the function away from that input without changing its value or slope at the contact point.

In fact, for any differentiable function $g$, the function

$$f(x)=mx+b+(x-a)^2g(x)$$

has the line $y=mx+b$ as its tangent at $x=a$. The added term is zero at $a$, so the function still has the line’s value there. Its derivative is

$$f'(x)=m+2(x-a)g(x)+(x-a)^2g'(x),$$

which gives $f'(a)=m$. Since $g$ can be chosen in many ways, this formula describes infinitely many functions with the same tangent line at the same point.

Worked example

Suppose $y=2x+1$ is tangent to a function at $x=3$. At the contact point, the line has height

$$2(3)+1=7.$$

Therefore the function must satisfy

$$f(3)=7, \qquad f'(3)=2.$$

One function that works is the line itself, $f(x)=2x+1$. But it is not the only choice. For example, take $g(x)=1$ in the general formula:

$$f(x)=2x+1+(x-3)^2.$$

At $x=3$, this function has value $7$. Its derivative is $f'(x)=2+2(x-3)$, so $f'(3)=2$. Thus it has the same tangent line, even though it is a different function. Choosing $g(x)=x$ or $g(x)=\sin x$ produces further examples.

How extra information can determine a function

You need enough additional conditions to narrow the possibilities to one function, along with a specified class of functions. For instance, if you are told that the function is linear, then its tangent line is the function itself: $f(x)=mx+b$. If you are told it is quadratic, a tangent condition at one point still leaves a free coefficient, so another independent condition is needed.

For example, if the function in the worked example is known to be a quadratic, write

$$f(x)=Ax^2+Bx+C.$$

The tangent conditions at $x=3$ give

$$9A+3B+C=7, \qquad 6A+B=2.$$

There are three unknown coefficients but only two equations. A further condition, such as another function value, is required to determine $A$, $B$, and $C$ uniquely. If you also know $f(0)=1$, then $C=1$; solving the equations gives $A=0$, $B=2$, and $f(x)=2x+1$.

Other useful additional information might include the value of the function at another point, a second derivative, a differential equation, or a condition such as “$f$ is a polynomial of degree at most two.” Whether the answer is unique depends on the kind and number of independent conditions, not just on having a tangent line.

Common mistakes

• Treating the tangent line as the function. The function may curve away from its tangent line immediately after the contact point. The line equals the function everywhere only if the function is known to be linear (or if the given information otherwise forces it).
• Using the slope but not the contact point. The line’s slope gives $f'(a)$, but you also need the contact point’s coordinates to obtain $f(a)$.
• Assuming any line is tangent at any chosen point. The line must pass through the graph’s point at that input, and its slope must match the derivative there.
• Confusing a tangent line with a secant line. A tangent describes the limiting slope at one point; a secant passes through two points on the graph.

If the tangent point’s $x$-coordinate is not given, the line alone usually provides even less information: you may not know where the function touches it. In either case, use the line to form the value-and-slope conditions, then check what additional information is available before claiming that the whole function is determined.