Answer:
The given function [tex]f(x)=\frac{2x}{3}-4[/tex] was replaced with [tex]f(x+k)=\frac{2x}{3}-6[/tex] then the value of k is -3
that is k=-3.
Step-by-step explanation:
The given function is [tex]f(x)=\frac{2x}{3}-4\hfill (1)[/tex]
and when x=x+k then [tex]f(x+k)=\frac{2x}{3}-6\hfill (2)[/tex]
Now to find the value of k:
Put x=x+k in the function f(x)
[tex]f(x)=\frac{2x}{3}-4[/tex]
[tex]f(x+k)=\frac{2(x+k)}{3}-4[/tex]
[tex]f(x+k)=\frac{2x+2k}{3}-4\hfill (3)[/tex]
Now comparing equations (2) and (3) we get
[tex]f(x+k)=\frac{2x+2k}{3}-4=\frac{2x}{3}-6[/tex]
[tex]\frac{2x+2k}{3}-4=\frac{2x}{3}-6[/tex]
[tex]\frac{2x+2k}{3}-4-\frac{2x}{3}+6=0[/tex]
[tex]\frac{2x}{3}+\frac{2k}{3}-4-\frac{2x}{3}+6=0[/tex]
[tex]\frac{2k}{3}+2=0[/tex]
[tex]\frac{2k}{3}=-2[/tex]
[tex]k=\frac{-2\times 3}{2}[/tex]
[tex]k=-3[/tex]
Therefore k=-3
Therefore the given function [tex]f(x)=\frac{2x}{3}-4[/tex] was replaced with [tex]f(x+k)=\frac{2x}{3}-6[/tex] then the value of k is -3
that is k=-3.
