Use the displayed table to determine the linear and exponential models, compare their values and gaps at every listed input, calculate both values at \(x~=~5\), and locate the crossing bracket and first listed integer where the exponential model overtakes the linear model.

A linear pattern can stay ahead for several inputs even while an exponential pattern is preparing to pass it. We’ll derive each rule from its starting value and repeated change, compare matching rows, and track the signed gap rather than relying on appearances. Extending both rules to the next input will reveal whether the order reverses and bracket the crossing between consecutive inputs.