Question

The equation represents the function f, and the graph represents the function g.f(x) = 6x - 5Determine the relationship between the rates of change of f and g

Answer 1

Given that;

The equation represents the function f, and the graph represents the function g.

Where

`f(x)=6x-5`

To find the slope of g(x), we will pick points on the graph

`\begin{gathered} (x_1,y_1)=(0,2) \\ (x_2,y_2)=(-1,-1) \end{gathered}`

The formula to find slope, m, is

`m=(y_2-y_1)/(x_2-x_1)`

Substitute the values of the variables into the formula above

`\begin{gathered} m=(-1-2)/(-1-0)=(-3)/(-1)=3 \\ m=3 \end{gathered}`

The slope of g(x) is 3

The slope of f(x) is 6,

Where,

Rate of change is the same as the slope of the function/graph

The relationship between the rates of change of f and g will be

`\begin{gathered} \frac{Rate\text{ of change of f\lparen x\rparen}}{Rate\text{ of change of g\lparen x\rparen}}=(6)/(3)=(2)/(1) \\ Rate\text{ of change of f\lparen x\rparen}:Rate\text{ of change of g\lparen x\rparen}=2:1 \end{gathered}`

Hence, the rate of change of f is twice the rate of change of g (option D)