A function F is defined on the set R, of real numbers by \(f : x \to px^{2} + qx + 2\), where p and q are constants. If \(f(-2) = 0\) and \(f(1) = 3\), find...
Question 1 Report
A function F is defined on the set R, of real numbers by \(f : x \to px^{2} + qx + 2\), where p and q are constants. If \(f(-2) = 0\) and \(f(1) = 3\), find \(f(-4)\).
Answer Details
Given \(f(x)=px^2+qx+2.\)
From \(f(-2)=0:\ p(-2)^2+q(-2)+2=0\Rightarrow 4p-2q+2=0\Rightarrow 2p-q=-1.\)
From \(f(1)=3:\ p+q+2=3\Rightarrow p+q=1.\)
Add the two results: \((2p-q)+(p+q)=-1+1\Rightarrow 3p=0\Rightarrow p=0.\) Then \(q=1.\)
So \(f(x)=x+2.\) Therefore
\[f(-4)=-4+2=-2.\]
Check: \(f(-2)=-2+2=0\) and \(f(1)=1+2=3,\) as required.
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