Knowledge Science · Comparing functions

1. Compare the values

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Example graph

f(x)=x^2,\quad g(x)=2x+1

At a shared input x, compare the heights of the parabola and the line. Their vertical separation is |f(x) − g(x)|. A small gap says the values are close; it does not tell us whether the slopes match.

f(1): 1
g(1): 3
Vertical separation: 2

The solid curve is f; the dashed curve is g. The vertical segment measures |f(x) − g(x)| at the selected input. The visible window is −5 ≤ x, y ≤ 5. Readouts are rounded to six decimal places.

2. Where curves intersect

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Example graph

f(x)=x^2,\quad g(x)=2x

For x² and 2x, the gap vanishes at x = 0 and x = 2. These are intersections. At x = 2 the slopes differ: f′(2) = 4 while g′(2) = 2, so the curves cross rather than share a tangent.

f(1): 1
g(1): 2
Vertical separation: 1

The solid curve is f; the dashed curve is g. The vertical segment measures |f(x) − g(x)| at the selected input. The visible window is −5 ≤ x, y ≤ 5. Readouts are rounded to six decimal places.

3. Tangent contact

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Example graph

f(x)=x^2,\quad g(x)=2x-1

The line g(x) = 2x − 1 is tangent to f(x) = x² at x = 1. Both have value 1 and slope 2 there. Their vertical gap is (x − 1)², so it shrinks as you approach the contact point.

f(2): 4
g(2): 3
Vertical separation: 1

The solid curve is f; the dashed curve is g. The vertical segment measures |f(x) − g(x)| at the selected input. The visible window is −5 ≤ x, y ≤ 5. Readouts are rounded to six decimal places.

4. A local approximation

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Example graph

f(x)=\sin(x),\quad g(x)=x

In radians, sin(x) and its tangent g(x) = x share value 0 and slope 1 at zero. Bring x toward zero to see their gap shrink. The line is useful nearby, but it does not replace the sine curve everywhere.

f(1): 0.841471
g(1): 1
Vertical separation: 0.158529

The solid curve is f; the dashed curve is g. The vertical segment measures |f(x) − g(x)| at the selected input. The visible window is −5 ≤ x, y ≤ 5. Readouts are rounded to six decimal places.