Question

answer the questions below to determine what kind of function is depicted in the table below.

x	1	2	3	4	5

answer attempt 2 out of 2
this function is none of the above because
none of the abo
quadratic
linear
exponential
none of the above
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Explanation:

Step1: Check for linearity

Find differences of $f(x)$ values.
$f(2)-f(1)=43 - (-3)=46$, $f(3)-f(2)=181 - 43 = 138$. Since $46
eq138$, not linear.

Step2: Check for quadraticity

Find second - order differences. First differences: $46,138,230,322$. Second differences: $138 - 46=92$, $230 - 138 = 92$, $322-230 = 92$. But for a quadratic function $y = ax^{2}+bx + c$, the second - order differences should be constant and the general form doesn't match easily by substituting points.

Step3: Check for exponentiality

Let $y = ab^{x}+c$. Substitute $(1,-3)$: $ab + c=-3$. Substitute $(2,43)$: $ab^{2}+c = 43$. Subtract first equation from second: $ab^{2}-ab=46$, $ab(b - 1)=46$. Substitute $(3,181)$: $ab^{3}+c=181$, subtract $ab^{2}+c = 43$ gives $ab^{3}-ab^{2}=138$, $ab^{2}(b - 1)=138$. Then $\frac{ab^{2}(b - 1)}{ab(b - 1)}=\frac{138}{46}=3$, so $b = 3$. Substituting $b = 3$ into $ab(b - 1)=46$ gives $6a=46$, $a=\frac{23}{3}$, and substituting $a=\frac{23}{3},b = 3$ into $ab + c=-3$ gives $23 + c=-3$, $c=-26$. But substituting other points doesn't hold exactly. So it is none of the above.

Answer:

none of the above