Question

The graphs of the functions f and g are shown. The function _____ (f/g) has a greater average rate of change between x = 0 and x = 1. The function ______ (f/g) has a greater average rate of change between x = 1 and x = 2. The functions f and g have the same average rate of change between x = ____and x = ______.

Answer 1

The function f has a greater average rate of change between x = 0 and x = 1.

The function g has a greater average rate of change between x = 1 and x = 2.

The functions f and g have the same average rate of change at some point between x = 0 and x = 1. That point is x = 3/8.

Answer 2

- The function
`\( g \)` has a greater average rate of change between
`\( x = 0 \)` and
`\( x = 1 \).`

- The second part cannot be determined without additional information or a clearer view of the graph.

- The function f and g have the same average rate of change between x =0 and x=1.5

The average rate of change of a function over an interval is the change in the function's value divided by the change in the input value (often the change in
`\( y \)` divided by the change in
`\( x \))` over that interval. It is essentially the slope of the secant line between two points on the function's graph.

1. Average rate of change between
`\( x = 0 \)` and
`\( x = 1 \)`:

- The function
`\( f \)` has a smaller slope than function
`\( g \)` between
`\( x = 0 \)` and
`\( x = 1 \)`. This can be observed because \( g \) increases more rapidly than \( f \) over this interval.

- Therefore, the function
`\( g \)` has a greater average rate of change between
`\( x = 0 \)` and
`\( x = 1 \).`

2. Average rate of change between
`\( x = 1 \)` and
`\( x = 2 \):`

- Without specific values, we can only make a qualitative analysis based on the steepness of the curve.

- If the trend that
`\( g \)` increases faster than
`\( f \)` continues beyond
`\( x = 1 \)`, then
`\( g \)` would still have a greater average rate of change between
`\( x = 1 \)` and
`\( x = 2 \)`. However, we cannot confirm this without more information or a clearer view of the functions beyond
`\( x = 1 \)`.

3. The same average rate of change:

both function f and g intersect at x =0 and x=1.5

Such that , f(1.5) = g(1.5)

and f(0) = g(0)

Hence the function f and g have the same average rate of change between x =0 and x=1.5

- The function
`\( g \)` has a greater average rate of change between
`\( x = 0 \)` and
`\( x = 1 \).`

- The second part cannot be determined without additional information or a clearer view of the graph.

- The function f and g have the same average rate of change between x =0 and x=1.5