choosing the graph of the sum of two functions\na moving sidewalk travels at a rate defined by the function…

choosing the graph of the sum of two functions\na moving sidewalk travels at a rate defined by the function $f(x)$. a boy on the sidewalk is walking at a rate defined by the function $g(x)$. the sum of the functions, $h(x)$, represents the rate he is moving relative to his surroundings. which could be the graph of the functions?

choosing the graph of the sum of two functions\na moving sidewalk travels at a rate defined by the function $f(x)$. a boy on the sidewalk is walking at a rate defined by the function $g(x)$. the sum of the functions, $h(x)$, represents the rate he is moving relative to his surroundings. which could be the graph of the functions?

Answer

Explanation:

Step1: Recall function sum property

For any $x$, $h(x)=f(x)+g(x)$, so $h(x)$ must be greater than both $f(x)$ and $g(x)$ for positive $x$ (since all rates are positive here).

Step2: Analyze each graph

  • Graph1: $h(x) > g(x) > f(x)$, fits $h=f+g$.
  • Graph2: $h(x) < f(x), g(x)$, violates $h=f+g$.
  • Graph3: $f(x) > h(x) > g(x)$, violates $h=f+g$.
  • Graph4: $h(x) > g(x) > f(x)$, but the slope difference does not match additive growth (the gap between $h$ and $g$ should equal $f$'s value, which is only true for the first graph).

Answer:

The first graph (leftmost one, with $h(x)$ as the steepest line above $g(x)$ and $f(x)$)