Problem

Compare \(f(x)~=~5x~+~2\) and \(g(x)~=~3x~+~10\) using their initial values and rates. Calculate the crossover point and state the function comparison on either side.

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What this problem is really about

A complete comparison of two lines should connect their starting gap, their relative rate, and their crossover. We’ll read the intercepts and slopes, determine how quickly the initial gap closes, and solve where the outputs match. The crossover then divides the number line into regions where the sign of the difference tells which function is greater.

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initials = f(0)=2 and g(0)=10, so g starts 8 higher; rates = f slope 5 and g slope 3, so f gains 2 more per input; crossover = x=4, point (4,22); comparison = g>f for x<4, equal at 4, f>g for x>4

initials = f(0)=2 and g(0)=10, so g starts 8 higher; rates = f slope 5 and g slope 3, so f gains 2 more per input; crossover = none; comparison = g>f for all x because it starts higher

initials = f(0)=2 and g(0)=10, so g starts 8 higher; rates = f slope 5 and g slope 3, so f gains 2 more per input; crossover = x=8, point (8,42); comparison = g>f for x<8, equal at 8, f>g for x>8

initials = f(0)=2 and g(0)=10, so f starts higher; rates = f slope 5 and g slope 3, so g gains 2 more per input; crossover = x=4, point (4,22); comparison = f>g for x<4, equal at 4, g>f for x>4

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Compare \(f(x)~=~5x~+~2\) and \(g(x)~=~3x~+~10\) using their initial values and rates. Calculate the crossover point and state the function comparison on either side.

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