Question

Below is the graph of f(x) and the linear function g(x).Question) compare the two function when x=3. Remember to show any work and write your final comparison in the form of a complete sentence.

Answer 1

From the f(x) plot below

the equation of f(x) can be evaluated by picking any two points on the plot.

The selected points are (0, 4) and (6, 0).

Thus, the equation of the line passing through these points is evaluated as

`\begin{gathered} y-y_1=m(x-x_1) \\ \text{where} \\ m=(y_2-y_1)/(x_2-x_1) \end{gathered}`

In this case,

`\begin{gathered} (x_1,y_1)\Rightarrow(0,4) \\ (x_2,y_2)\Rightarrow(6,0) \end{gathered}`

Thus,

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1)\text{ = }(0-4)/(6-0)=-(4)/(6) \\ m=-(2)/(3) \\ \text{thus,} \\ y-y_1=m\mleft(x-x_1\mright) \\ y-4=-(2)/(3)(x-0) \\ y-4=-(2)/(3)x \\ y=-(2)/(3)x+4 \end{gathered}`

Thus, the f(x) function is expressed as

`f(x)\text{ = -}(2)/(3)x+4`

Given that

`g(x)\text{ = }-4x+11`

When x=3,

`\begin{gathered} f(x)\text{ = -}(2)/(3)x+4 \\ \text{substitute the value of 3 for x into the f(x) function} \\ f(x)\text{ = -}(2)/(3)(3)+4=-2+4=2 \end{gathered}`
`\begin{gathered} g(x)\text{ = }-4x+11 \\ \text{substitute the value of 3 for x into the g(x) function} \\ g(3)\text{ = -4(3)+11=-12+11=-1} \end{gathered}`

Thus, when x = 3, the values of f(x) and g(x) are 2 and -1 respectively.