Problem 9 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 9-14, y=f(x) .\) Find \(f(-3)\) for each function. \(f(x)=7-x\)

Step-by-Step Solution

Answer

\(f(-3) = 10\)

Step 1: Substitute the Value of x

To find \(f(-3)\) for the function \(f(x) = 7 - x\), we substitute \(-3\) into the function in place of \(x\). This gives us \(f(-3) = 7 - (-3)\).

Step 2: Simplify the Expression

Simplify the expression \(7 - (-3)\). Remember that subtracting a negative is the same as addition, so this becomes \(7 + 3\).

Step 3: Calculate the Result

Now, perform the addition: \(7 + 3 = 10\).

Step 4: State the Final Answer

The final result is \(f(-3) = 10\).

Key Concepts

Substitution Method

The substitution method is a technique used to evaluate functions by replacing the variable with a given number. This makes it easier to calculate and understand real-world problems involving functions. Here’s how it works:

• First, identify which variable in the function will be replaced. Typically, this is represented by the letter \(x\) in the function \(f(x)\).
• The next step is to substitute the value you have been given—in this case, \(-3\). Replace each occurrence of \(x\) in the function rule with this value.
• Once the substitution is made, proceed with any calculations necessary to find the outcome.

By substituting \(-3\) into the function \(f(x) = 7 - x\), you transform the expression into \(7 - (-3)\), which is then simplified to complete the calculation. This method streamlines the process of finding specific values of function outputs.

Function Notation

Function notation is a convenient, clear, and precise way to express the relationship between input and output values in mathematical functions. It uses the format \(f(x)\) to denote a function named "\(f\)" with "\(x\)" as the input variable.

• The letter \(f\) does not have to be \(f\); other letters can be used to represent different functions, such as \(g(x)\), \(h(x)\), etc.
• The expression inside the parentheses, "\(x\)", is the input value for the function. The entire function notation, \(f(-3)\), means you need to evaluate the function for \(x = -3\).
• When the input \(x\) is replaced by \(-3\), as in \(f(-3)\), it communicates clearly that you are evaluating the function at that specific point.

Function notation is particularly useful because it instantly communicates what input you are using and can help keep track of how the function behaves at different values of \(x\). This clarity is essential in both complex mathematics and practical applications.

Simplifying Expressions

Simplifying expressions is a key step in solving problems involving functions. This process involves combining terms, using mathematical operations to make calculations easier, and making the expression more manageable.

• In the context of substituting values into functions, begin by conducting arithmetic operations such as addition, subtraction, multiplication, or division as required.
• In the example \(7 - (-3)\), remember that subtracting a negative is equivalent to adding a positive. Hence, \(7 - (-3)\) translates to \(7 + 3\).
• Carefully perform each operation step-by-step to ensure no mistakes in calculation.

After simplifying the expression, the final calculation yields \(10\) for \(f(-3)\). Simplification is crucial because it clears the clutter in expressions and makes it easier to find the correct result.