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Cora is using successive approximations to estimate a positive solution to
f(x)=g(x), where
f(x)=x^(2)+13 and
g(x)=3x+14. The table shows her results for different input values of
x.

x

f(x)

g(x)

0
13
14

1
14
17

2
17
20

3
22
23

4
29
26

3.5
25.25
24.5

$4$ . Cora is using successive approximations to estimate a positive solution to $f(x)=g(x)$ , where $f(x)=x^{2}+13$ and $g(x)=3 x+14$ . The table shows her results for different input values of $x$ . $\newline$ \begin{tabular}{|l|l|l|} $\newline$ \hline $x$ & $f(x)$ & $g(x)$ \\ $\newline$ \hline $0$ & $13$ & $14$ \\ $\newline$ \hline $1$ & $14$ & $17$ \\ $\newline$ \hline $2$ & $17$ & $20$ \\ $\newline$ \hline $3$ & $22$ & $23$ \\ $\newline$ \hline $4$ & $29$ & $26$ \\ $\newline$ \hline $3$ . $5$ & $25$ . $25$ & $24$ . $5$ \\ $\newline$ \hline $\newline$ \end{tabular}

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Transformations of functions

Full solution

Q.

4

. Cora is using successive approximations to estimate a positive solution to

f(x)=g(x)

, where

f(x)=x^{2}+13

and

g(x)=3 x+14

. The table shows her results for different input values of

x

.

\newline

\begin{tabular}{|l|l|l|}

\newline

\hline

x

&

f(x)

&

g(x)

\\

\newline

\hline

0

&

13

&

14

\\

\newline

\hline

1

&

14

&

17

\\

\newline

\hline

2

&

17

&

20

\\

\newline

\hline

3

&

22

&

23

\\

\newline

\hline

4

&

29

&

26

\\

\newline

\hline

3

.

5

&

25

.

25

&

24

.

5

\\

\newline

\hline

\newline

\end{tabular}

Calculate $f(x)$ and $g(x)$ for $x = 0$ : Calculate $f(x)$ and $g(x)$ for $x = 0$ : $\newline$ $f(0) = 0^2 + 13 = 13$ $\newline$ $g(0) = 3\times0 + 14 = 14$
Calculate $f(x)$ and $g(x)$ for $x = 1$ : Calculate $f(x)$ and $g(x)$ for $x = 1$ : $\newline$ $f(1) = 1^2 + 13 = 14$ $\newline$ $g(1) = 3 \times 1 + 14 = 17$
Calculate $f(x)$ and $g(x)$ for $x = 2$ : Calculate $f(x)$ and $g(x)$ for $x = 2$ : $\newline$ $f(2) = 2^2 + 13 = 17$ $\newline$ $g(2) = 3 \times 2 + 14 = 20$
Calculate $f(x)$ and $g(x)$ for $x = 3$ : Calculate $f(x)$ and $g(x)$ for $x = 3$ : $f(3) = 3^2 + 13 = 22$ $g(3) = 3\times3 + 14 = 23$
Calculate $f(x)$ and $g(x)$ for $x = 4$ : Calculate $f(x)$ and $g(x)$ for $x = 4$ : $\newline$ $f(4) = 4^2 + 13 = 29$ $\newline$ $g(4) = 3 \times 4 + 14 = 26$
Calculate $f(x)$ and $g(x)$ for $x = 3.5$ : Calculate $f(x)$ and $g(x)$ for $x = 3.5$ : $\newline$ $f(3.5) = 3.5^2 + 13 = 25.25$ $\newline$ $g(3.5) = 3\times3.5 + 14 = 24.5$
Analyze the results to find where $f(x)$ and $g(x)$ are closest: Analyze the results to find where $f(x)$ and $g(x)$ are closest: $\newline$ At $x = 3$ , $f(3) = 22$ and $g(3) = 23$ $\newline$ At $x = 3.5$ , $f(3.5) = 25.25$ and $g(3.5) = 24.5$ $\newline$ The values are closest at $x = 3.5$ , but they do not exactly match.

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