Question

Consider functions fand gUsing a table of values, what is the approximate solution to the equation f(I) = g(I| to the nearest quarter of a unit?

Answer 1

The correct option is c.

The approximate solution to the equation
`$f(x) = g(x)$` to the nearest quarter of a unit is
`$x \approx 0.75$`

To approximate the solution to the equation
`$f(x) = g(x)$` to the nearest quarter of a unit using a table of values, we'll first calculate the values of
`$f(x)$` and
`$g(x)$` for a few different values of
`$x$` and then find the value of
`$x$` for which these two functions are closest to each other. Here are the steps:

1. Calculate the values of
`$f(x)$` and
`$g(x)$` for some values of
`$x$`.

We'll choose a range of values for
`$x$` and calculate both
`$f(x)$` and
`$g(x)$`:

For
`$x = -2.25$`:

`$f(-2.25)=(-2.25-1)/((-2.25)^2+(-2.25)-1) \approx 0.683 g(-2.25)=3^2-2=7$`

For
`$x = -1.75$`:

`$f(-1.75)=(-1.75-1)/((-1.75)^2+(-1.75)-1) \approx-0.276 g(-1.75)=3^2-2=7$`

For
`$x = 0.50$`:

`$f(0.50)=(0.50-1)/((0.50)^2+(0.50)-1) \approx 0.732 g(0.50)=3^2-2=7$`

For
`$x = 0.75$`:

`$f(0.75)=(0.75-1)/((0.75)^2+(0.75)-1) \approx 0.889 g(0.75)=3^2-2=7$`

2. Compare the values of
`$f(x)$` and
`$g(x)$` to find the closest match.

From the calculations above, we can see that the values of
`$f(x)$` and
`$g(x)$` are closest for
`$x = 0.75$`.

So, The answer is x=0.75.

Answer 2

Given:

Two functions f(x) and g(x) are given.

`\begin{gathered} f(x)=(x-1)/(x^2+x-1) \\ g(x)=3^x-2 \end{gathered}`

Required:

Find the approximate solution to the equation f(x)=g(x) to the nearest quarter of the unit.

Step-by-step explanation:

(a) substitute x= 0.50in f(x) and g(x).

`\begin{gathered} f(x)=(0.5-1)/((0.5)^2+0.5-1) \\ f(x)=(-0.5)/(-0.25) \\ f(x)=2 \end{gathered}`
`\begin{gathered} g(x)=3^(0.5)-2 \\ g(x)=-0.268 \end{gathered}`

(b) substitute x=-1.75 in f(x) and g(x).

`\begin{gathered} f(x)=(-1.75-1)/((-1.75)^2-1.75-1) \\ f(x)=(-2.75)/(0.38125) \\ f(x)=-7.2 \end{gathered}`
`\begin{gathered} g(x)=3^(-1.75)-2 \\ g(x)=-1.854 \end{gathered}`

(c) substitute x= 0.75 in f(x) and g(x).

`\begin{gathered} f(x)=(0.75-1)/((0.75)^2+0.75-1) \\ f(x)=(-0.25)/(0.3125) \\ f(x)=-0.8 \end{gathered}`
`\begin{gathered} g(x)=3^(0.75)-2 \\ g(x)=2.795 \end{gathered}`

(d) substitute x= -2.25 in f(x) and g(x).

`\begin{gathered} f(x)=(-2.25-1)/((-2.25)^2-2.25-1) \\ f(x)=(-3.25)/(1.8125) \\ f(x)=-1.793 \end{gathered}`
`\begin{gathered} g(x)=3^(-2.25)-2 \\ g(x)=-1.916 \end{gathered}`

we can observe that at x = -2.25 the functions f(x) and g(x) has the approximate values.

Final Answ