Problem 27 - Elementary and Intermediate Algebra

Question

To prepare for Section \(9.5,\) review functions (Sections 3.8 and \(5.9)\) Find each of the following, given \(f(x)=80 x+2500\) and \(g(x)=150 x\) $$ f(90) \quad[3.8] $$

Step-by-Step Solution

Answer

f(90) = 9700

Step 1: Identify the Function

Given the function: \[ f(x) = 80x + 2500 \]

Step 2: Substitute the Value

To find \(f(90)\), substitute \(x = 90\) into the function: \[ f(90) = 80(90) + 2500 \]

Step 3: Perform the Multiplication

Multiply 80 by 90: \[ 80 \times 90 = 7200 \]

Step 4: Add the Constant Term

Add the result to 2500: \[ 7200 + 2500 = 9700 \]

Step 5: Conclusion

Thus, \( f(90) = 9700 \).

Key Concepts

Substitution

In mathematics, substitution is the process of replacing a variable with a given number. This is useful when we want to evaluate a function at a specific point. In our example, we have the function \( f(x) = 80x + 2500 \). To find \( f(90) \), we substitute 90 for every instance of \( x \) in the function. This transforms our function from \( f(x) = 80x + 2500 \) to \( f(90) = 80(90) + 2500 \).

Multiplication

Once we have substituted the value, the next step is often multiplication, especially when dealing with linear functions like \( f(x) = 80x + 2500 \). In our example, we need to multiply 80 by 90. This can be calculated as follows: \( 80 \times 90 = 7200 \). Multiplication helps us scale the variable according to the function's formula. Remember to perform multiplication before addition when following order of operations (PEMDAS/BODMAS).

Addition

After substituting the value and performing the multiplication, the final step is addition. In our function \( f(x) = 80x + 2500 \), after calculating \( 80 \times 90 = 7200 \), we add 2500 to the result. Thus, we have \( 7200 + 2500 = 9700 \). Addition combines the results to give the final value of the function at that specific point. The final result in our example is \( f(90) = 9700 \).