Answer:
a) For h(2)=5 then k=4
b) For h(3)=k then k=3
c) For h(k)=k then k=3
Step-by-step explanation:
Given that the function h is defined by
[tex]h(x)=\frac{kx-3}{x-1}[/tex] where [tex]x\neq1[/tex]
a) To find for h(2)=5:
[tex]h(x)=\frac{kx-3}{x-1}[/tex] where [tex]x\neq1[/tex]
that is put x=2 in the above function
[tex]h(2)=\frac{k(2)-3}{2-1}=5[/tex]
Now [tex]\frac{2k-3}{1}=5[/tex]
[tex]2k-3=5[/tex]
[tex]2k=5+3[/tex]
[tex]k=\frac{8}{2}[/tex]
k=4
Therefore k=4
b) To find for h(3)=k:
[tex]h(x)=\frac{kx-3}{x-1}[/tex] where [tex]x\neq1[/tex]
that is put x=3 in the above function
[tex]h(3)=\frac{k(3)-3}{3-1}=k[/tex]
Now [tex]\frac{3k-3}{2}=k[/tex]
[tex]3k-3=2k[/tex]
[tex]3k-2k-3=0[/tex]
[tex]k-3=0[/tex]
k=3
Therefore k=3
c) h(k)=k
To find for h(3)=K:
[tex]h(x)=\frac{kx-3}{x-1}[/tex] where [tex]x\neq1[/tex]
that is put x=k in the above function
[tex]h(k)=\frac{k(k)-3}{k-1}=k[/tex] here [tex]k\neq1[/tex]
Now [tex]\frac{k^2-3}{k-1}=k[/tex]
[tex]k^2-3=k(k-1)[/tex]
[tex]k^2-3=k^2-k[/tex]
[tex]k^2-3-k^2+k=0[/tex]
k-3=0
k=3
Therefore k=3
