Full solution

Q. Three points on the graph of the function

f(x)

are

\{(0,3),(1,9),(2,27)\}

. Which equation represents

f(x)

\newline

f(x)=6 x+3

\newline

f(x)=3(3)^{x}

\newline

f(x)=18 x-9

\newline

f(x)=x^{2}+3

Given Points Testing: We are given three points on the graph of the function $f(x)$ : $(0,3)$ , $(1,9)$ , and $(2,27)$ . To determine which equation represents $f(x)$ , we can plug these points into each of the given equations to see which one is consistent with all three points.
Equation $1$ Testing: First, let's test the point $(0,3)$ with the equation $f(x)=6x+3$ . $\newline$ $f(0) = 6\times 0 + 3 = 3$ . $\newline$ This matches the given point $(0,3)$ , so this equation could be correct.
Equation $2$ Testing: Next, let's test the point $(1,9)$ with the equation $f(x)=6x+3$ . $f(1) = 6\times1 + 3 = 9$ . This also matches the given point $(1,9)$ , so this equation is still a candidate.
Equation $3$ Testing: Now, let's test the point $(2,27)$ with the equation $f(x)=6x+3$ . $f(2) = 6\times2 + 3 = 15$ . This does not match the given point $(2,27)$ , so $f(x)=6x+3$ cannot be the correct equation.
Final Equation Determination: Let's move on to the next equation, $f(x)=3(3)^{x}$ , and test the point $(0,3)$ . $f(0) = 3\cdot(3^{0}) = 3\cdot1 = 3$ .This matches the given point $(0,3)$ , so this equation could be correct.
Final Equation Determination: Let's move on to the next equation, $f(x)=3(3)^{x}$ , and test the point $(0,3)$ . $f(0) = 3\cdot(3^0) = 3\cdot1 = 3$ .This matches the given point $(0,3)$ , so this equation could be correct.Now, let's test the point $(1,9)$ with the equation $f(x)=3(3)^{x}$ . $f(1) = 3\cdot(3^1) = 3\cdot3 = 9$ .This matches the given point $(1,9)$ , so this equation is still a candidate.
Final Equation Determination: Let's move on to the next equation, $f(x)=3(3)^{x}$ , and test the point $(0,3)$ . $f(0) = 3\cdot(3^0) = 3\cdot1 = 3$ .This matches the given point $(0,3)$ , so this equation could be correct.Now, let's test the point $(1,9)$ with the equation $f(x)=3(3)^{x}$ . $f(1) = 3\cdot(3^1) = 3\cdot3 = 9$ .This matches the given point $(1,9)$ , so this equation is still a candidate.Next, let's test the point $(2,27)$ with the equation $f(x)=3(3)^{x}$ . $(0,3)$ $0$ .This matches the given point $(2,27)$ , so $f(x)=3(3)^{x}$ is the correct equation.
Final Equation Determination: Let's move on to the next equation, $f(x)=3(3)^{x}$ , and test the point $(0,3)$ . $f(0) = 3\cdot(3^0) = 3\cdot1 = 3$ .This matches the given point $(0,3)$ , so this equation could be correct.Now, let's test the point $(1,9)$ with the equation $f(x)=3(3)^{x}$ . $f(1) = 3\cdot(3^1) = 3\cdot3 = 9$ .This matches the given point $(1,9)$ , so this equation is still a candidate.Next, let's test the point $(2,27)$ with the equation $f(x)=3(3)^{x}$ . $(0,3)$ $0$ .This matches the given point $(2,27)$ , so $f(x)=3(3)^{x}$ is the correct equation.We do not need to test the remaining equations because we have already found the equation that matches all three given points. The correct equation is $f(x)=3(3)^{x}$ .