The function \( f \) is linear. If \( f(2) = 2 \) and \( f(-6) = 6 \), find \( f(6) \).
Answer (2)
The\ slope-intercept\ form:y=mx+b\\\\f(2)=2\to for\ x=2\to y=2\\f(-6)=6\to for x=-6\to y=6\\\\put\ in\ equation\ of\ the\ line:\\\\ -\left\{\begin{array}{ccc}2=2m+b\\6=-6m+b\end{array}\right\ \ \ |subtract\ sides\ of\ the\ equations\\----------\\.\ \ \ -4=8m\ \ \ \ \ |divide\ both\ sides\ by\ 8\\.\ \ \ \ \boxed{m=-\frac{1}{2}}\\\\put\ m=-\frac{1}{2}\ to\ equation\ 2=2m+b:\\\\2=2\cdot(-\frac{1}{2})+b\\2=-1+b\ \ \ \ |add\ 1\ to\ both\ sides\\\boxed{b=3}\\\\\boxed{\boxed{y=-\frac{1}{2}x+3}}
t h ere f ore : f ( x ) = − 2 1 x + 3 ⇒ f ( 6 ) = − 2 1 ⋅ 6 + 3 = − 3 + 3 = 0 A n s w er : f ( 6 ) = 0
The linear function f ( x ) = − 2 1 x + 3 is derived from the given points. The value of the function at x = 6 is f ( 6 ) = 0 .
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