Robustness Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

If an input

x = 10

has a measurement error of

\pm 1

, find the range of

f(x) = 2x+3

and assess whether

f

is robust to this error.

1
$x$ ranges from $9$ to $11$ .
2
$f(9) = 21$ , $f(11) = 25$ . So $f(x) \in [21, 25]$ .
3
The error in $x$ ( $\pm 1$ , i.e., 10%) translates to an error in $f$ of $\pm 2$ (about $\pm 9\%$ ). The function is robust — errors do not amplify significantly.

Answer

f(x) \in [21,25];\text{ robust (error ratio } \approx 1:1\text{)}

For a linear function

f(x)=2x+3

, an error

\Delta x

x

produces an error

2\Delta x

f

. The slope 2 is the amplification factor. Linear functions with moderate slopes are generally robust.

About Robustness

The property of a result, algorithm, or model remaining valid or approximately correct even when its assumptions are slightly violated.

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