Answer :
We will have the following:
a) f(x) is a linear function:
We find the slope using the two points given:
[tex]m=\frac{70-10}{2-1}\Rightarrow m=60[/tex]Now, we replace in the general formula for a linear function:
[tex]y-y_1=m(x-x_1)\Rightarrow y-10=(60)(x-1)[/tex][tex]\Rightarrow y-10=60x-60\Rightarrow y=60x-50[/tex]b) f(x) is a power function:
We know taht a power function is given by the general expression:
[tex]f(x)=kx^n[/tex]Now, we will have that for each point the following is true:
[tex]\begin{cases}10=k(1)^n \\ \\ 70=k(2)^n\end{cases}[/tex]Now, we solve both for k and equal them:
[tex]\begin{cases}k=\frac{10}{1^n} \\ \\ k=\frac{70}{2^n}\end{cases}\Rightarrow\frac{10}{1^n}=\frac{70}{2^n}\Rightarrow10\cdot2^n=70\cdot1^n[/tex][tex]\Rightarrow\ln (10\cdot2^n)=\ln (70\cdot1^n)\Rightarrow\ln (10)+\ln (2^n)=\ln (70)+\ln (1^n)[/tex][tex]\Rightarrow\ln (10)+n\ln (2)=\ln (70)+n\ln (1)\colon\ln (1)=0[/tex]So:
[tex]\ln (10)+n\ln (2)=\ln (70)+n\ln (1)\Rightarrow\ln (10)+n\ln (2)=\ln (70)[/tex][tex]\Rightarrow n\ln (2)=\ln (70)-\ln (10)\Rightarrow n\ln (2)=\ln (70/10)[/tex][tex]\Rightarrow n=\frac{\ln (7)}{\ln (2)}[/tex]Now, we replace this value in one of the expressions and solve for k:
[tex]10=k(1)^{\ln (7)/\ln (2)}\Rightarrow k=10[/tex][1 at any power is also 1], now we write the expression that would be described:
[tex]f(x)=10x^{\ln (7)/\ln (2)}[/tex]c) We remember that the general form of a exponential function is givven by:
[tex]f(x)=a\cdot b^x[/tex]Now, using this and the two points we calculate:
[tex]\begin{cases}10=a\cdot b^1 \\ \\ 70=a\cdot b^2\end{cases}[/tex]Now, we solve both for a and equal them:
[tex]\Rightarrow\begin{cases}a=\frac{10}{b} \\ \\ a=\frac{70}{b^2}\end{cases}\Rightarrow\frac{10}{b}=\frac{70}{b^2}\Rightarrow10b^2=70b[/tex][tex]\Rightarrow10b^2-70b=0\Rightarrow b=\frac{-(-70)\pm\sqrt[]{(-70)^2-4(10)(0)}}{2(10)}[/tex][tex]\Rightarrow\begin{cases}b=0 \\ \\ b=7\end{cases}[/tex]Now, since having a value of "b" equal 0 would make little sense, we work with b = 7. Then we replace in one of the expressions and solve for a, that is:
[tex]\Rightarrow10=a\cdot7^1\Rightarrow7a=10\Rightarrow a=\frac{10}{7}[/tex]From this, we will have that the exponential form would be:
[tex]f(x)=\frac{10}{7}(7)^x[/tex]