The given functions are:
[tex]\begin{gathered} f(x)=-6x+2 \\ g(x)=-2^{x+2}+6 \end{gathered}[/tex]
It is required to graph the functions, and then find the solutions of the equation:
[tex]-6x+2=-2^{x+2}+6[/tex]
To find the solution to the equation, it is sufficient to find the points of intersection of the drawn graphs. The x-coordinate of these points of intersection is the required solution.
We will start by using points to graph the equations:
Find some values of the function f(x)=-6x+2.
For x=0, substitute x=0 into the function:
[tex]f(0)=-6(0)+2=0+2=2[/tex]
It follows that a point on the graph is (0,2).
For x=1, substitute x=1 into the function:
[tex]f(1)=-6(1)+2=-6+2=-4[/tex]
Hence, another point on the graph of f(x) is (1,-4).
For x=2, substitute x=2 into the function:
[tex]f(2)=-6(2)+2=-12+2=-10[/tex]
Hence, a third point on the graph is (2,-10).
Plot these points on a graph as shown:
Join these points to draw the graph of f(x):
Using the same procedure as in the first function, that is finding points for some x-values, the graph of g(x) is shown in purple:
Locate the points of intersection of the graphs of f(x) and g(x):
The points of intersection labeled are (0,2) and (2,-10).
The x-coordinates of the points (0,2) and (2,-10) are the solutions of the equation, that is, x=0, x=2.
The solutions of the given equation are x=0 and x=2.
