- Substitute \(x = 7\) into the function \(g(x) = \frac{7}{8}x - \frac{1}{2}\).
- Calculate the first term: \(\frac{7}{8} \times 7 = \frac{49}{8}\).
- Find a common denominator to subtract \(\frac{1}{2}\), which is \(\frac{4}{8}\).
- The final result is \(\frac{49}{8} - \frac{4}{8} = \frac{45}{8}\), so the answer is \(\boxed{\frac{45}{8}}\).
Explanation
- Understanding the Problem
The problem asks to evaluate the function \(g(x) = \frac{7}{8}x - \frac{1}{2}\) at the point \(x = 7\). Evaluating a function at a specific point involves substituting the value of the independent variable, \(x\), with the given number.
- Substituting the Value of x
Substitute \(x = 7\) into the function expression:
\[g(7) = \frac{7}{8}(7) - \frac{1}{2}\]
First, perform the multiplication for the first term:
\[\frac{7}{8} \times 7 = \frac{49}{8}\]
So the expression becomes:
\[g(7) = \frac{49}{8} - \frac{1}{2}\]
- Performing the Calculation
To perform the subtraction \(\frac{49}{8} - \frac{1}{2}\), we need to find a common denominator. The least common multiple of 8 and 2 is 8.
We convert \(\frac{1}{2}\) to have a denominator of 8:
\[\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}\]
Now subtract the two fractions:
\[g(7) = \frac{49}{8} - \frac{4}{8} = \frac{49 - 4}{8} = \frac{45}{8}\]
- Conclusion
Comparing the calculated value \(\frac{45}{8}\) with the given options:
- \(\frac{45}{8}\)
- \(\frac{53}{8}\)
- \(\frac{49}{8}\)
- \(\frac{60}{7}\)
The correct option is \(\frac{45}{8}\). Therefore, the final result is \(\boxed{\frac{45}{8}}\).
Examples
In everyday life, functions like this are used to model proportional relationships with a starting offset. For example, if you are calculating the total cost of a service that charges a flat fee (like a delivery base fee) plus a rate per item, you might represent the total cost as a function \(f(n) = rn + b\), where \(r\) is the rate per item, \(n\) is the number of items, and \(b\) is the base fee. Evaluating this function for a specific number of items helps you determine the total cost before finalizing a purchase.