$$ \left. \begin{array} { l } { x ightarrow y } \ { 0 ightarrow 3 } \ { 1 ightarrow 5 } \ { 2 ightarrow 11 } \ { 3 ightarrow 21 } \ { 4 ightarrow 35 } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { x ightarrow y } \ { 0 ightarrow 3 } \ { 1 ightarrow 5 } \ { 2 ightarrow 11 } \ { 3 ightarrow 21 } \ { 4 ightarrow 35 } \end{array} ight. $$

UpStudy AI · Accepted Answer

We are given the pairs:
  0 → 3  
  1 → 5  
  2 → 11  
  3 → 21  
  4 → 35

Notice that f(0) = 3, f(1) = 5, f(2) = 11, f(3) = 21, and f(4) = 35. We suspect that the function might be quadratic and try to find a function of the form

f(x) = ax² + bx + c.

Step 1. Find c:
  f(0) = a·0² + b·0 + c = c = 3. So, c = 3.

Step 2. Use f(1):
  f(1) = a·1² + b·1 + 3 = a + b + 3 = 5  
  => a + b = 2    (Equation 1)

Step 3. Use f(2):
  f(2) = a·2² + b·2 + 3 = 4a + 2b + 3 = 11  
  => 4a + 2b = 8   (Subtract 3 from both sides)  
  Divide both sides by 2: 2a + b = 4   (Equation 2)

Step 4. Solve Equations 1 and 2:
  Subtract Equation 1 from Equation 2:  
  (2a + b) − (a + b) = 4 − 2  
  => a = 2.

Now substitute a = 2 into Equation 1:
  2 + b = 2  => b = 0.

Thus, the quadratic function is:
  f(x) = 2x² + 0·x + 3 = 2x² + 3.

Verification:
  f(3) = 2(3)² + 3 = 18 + 3 = 21  
  f(4) = 2(4)² + 3 = 32 + 3 = 35

The function correctly maps all the given x-values to their corresponding y-values.

Answer: y = 2x² + 3.
The function is $$ y = 2x^2 + 3 $$.

\( \left. \begin{array} { l } { x \rightarrow y } \\ { 0 \rightarrow 3 } \\ { 1 \rightarrow 5 } \\ { 2 \rightarrow 11 } \\ { 3 \rightarrow 21 } \\ { 4 \rightarrow 35 } \end{array} \right. \)

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