Question

Alex and Joy are friends. They both had $\$100$ each initially. Alex earned $\$2$ on $1st$ day, $\$4$ on $2nd$ day, $\$8$ on $3rd$ day, $\$16$ on $4th$ day, and so on, while Joy earned $\$20$ everyday.
The function between the number of days and total earnings of Joy will be a __________ function and that of Alex will be a ___________ function.

• A. linear; nonlinear
• B. nonlinear; linear
• C. linear; linear
• D. nonlinear; nonlinear

Answer 1

The correct option is A linear; nonlinear

We will analyze both the functions between the number of days and the total money with them.
Let us consider $x=$ the number of days they will earn
And $y=$ total money one will earn in x days
We know the standard form of linear function is $y=mx+b$.

$\to$ Let us consider the earnings of Alex.
Initially Alex had $\$100$.
Alex earned $\$2$ on 1st day, $\$4$ on 2nd day, $\$8$ on 3rd day, $\$16$ on 4th day, and so on.
We can say Alex earned $\$21$ on 1st day, $\$22$ on 2nd day, $\$23$ on 3rd day, $\$24$ on the 4th day, and so on.
So after $x$ days, total earnings by Alex will be $=y=100+21+22+23+24+\dots \dots \dots +2x$
$\Rightarrow y=(21+22+23+24+\dots \dots \dots +2x)+100$
Here, the above equation cannot be expressed as $y=mx+b$.
The function between the number of days and total earnings of Alex will be a nonlinear function.

$\to$ Let us consider the earnings of Joy.
Initially Joy had $\$100$.
Joy earned $\$20$ everyday.
So after x days, total earnings by Joy will be $=y=100+20+20+\dots \dots \dots +$ up to $x$ days
$\Rightarrow y=100+20x$
$\Rightarrow y=20x+100$
Here, the above expression can be expressed as $y=mx+b$.
The function between the number of days and total earnings of Joy will be a linear function.

So option A is the correct answer.