If $f(x)$ varies directly with $x$, and $f(x)=56$ when $x=8$, find the value of $f(x)$ when $x=2$.
We are given that a function \(f(x)\) varies directly with \(x\). By definition, a function that varies directly with \(x\) can be represented by the equation \(f(x) = k \cdot x\), where \(k\) is a non-zero constant known as the constant of variation. We are also given the specific condition that \(f(8) = 56\). Our goal is to determine the value of \(f(2)\).
Using the direct variation formula \(f(x) = k \cdot x\), we substitute the known values \(f(8) = 56\) and \(x = 8\):
\[56 = k \cdot 8\]
To solve for the constant \(k\), we divide both sides by 8:
\[k = \frac{56}{8} = 7\]
Thus, the specific relationship for this function is \(f(x) = 7x\).
Now that we have the function \(f(x) = 7x\), we can easily find the value of \(f(x)\) for any \(x\). To find \(f(2)\), we substitute \(x = 2\) into our derived formula:
\[f(2) = 7 \cdot 2 = 14\]
Therefore, the value of \(f(2)\) is 14.
Direct variation is a fundamental concept used in many everyday situations. For example, consider the cost of purchasing gasoline. If the price per gallon is fixed, the total cost \(C\) varies directly with the number of gallons \(g\) purchased (\(C = k \cdot g\)). If you know that 8 gallons cost \(\$56\), you can determine that the price per gallon is \(\$7\). Using this constant of variation, you can easily calculate that 2 gallons will cost \(2 \times \$7 = \$14\). This principle of proportional relationships is used constantly in economics, engineering, and personal finance to predict costs and measure outcomes.