Problem 31 - Algebra A Combined Function

Question

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ f(1) $$

Step-by-Step Solution

Answer

f(1) = 6.

Step 1: Identify the Function

The function given is \( f(x) = \frac{x^2 + 5}{x} \). We need to find \( f(1) \).

Step 2: Substitute the Value

Substitute \( x = 1 \) into the function \( f(x) \). This gives us \( f(1) = \frac{1^2 + 5}{1} \).

Step 3: Simplify the Expression

Simplify the expression \( \frac{1^2 + 5}{1} \). This becomes \( \frac{1 + 5}{1} \).

Step 4: Calculate the Result

Perform the arithmetic to find that \( \frac{1 + 5}{1} = 6 \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra and calculus to solve equations or find specific function values. It involves replacing variables with numerical values or expressions. When you substitute, you're essentially "plugging in" a number wherever a variable appears in a function.

Here's how you do it:

• Identify the variable you need to substitute. In our problem, it's the variable \(x\).
• Replace every instance of the variable in the function with the given number. For \(f(x)=\frac{x^2 + 5}{x}\), substituting \(x = 1\) means replacing both \(x\) terms with \(1\).
• You end up with a new expression that only contains numbers. This makes it simpler to calculate the function's value.

This method is often used because it allows for a straightforward evaluation of complex functions.

Simplifying Expressions

Simplification is all about reducing an expression to its simplest form. This makes it easier to understand or compute. After substituting values into the function, the next step is to simplify the resulting mathematical expression.

For example, with \(f(1) = \frac{1^2 + 5}{1}\), your expression is initially \(\frac{1 + 5}{1}\). To simplify:

• Perform arithmetic operations like addition or subtraction inside the parentheses first.
• Break down the fractions by dividing the numerator by the denominator when applicable.
• Ensure the simplest form by canceling common terms if any, although in this example all calculations are straightforward.

This simplification makes the calculation process quick and yields the solution with minimal steps, resulting in an answer that's easy to comprehend.

Evaluating Functions

Evaluating functions means finding the output or value of a function for a specific input. It's like asking, "What does this function equal when I insert this number?"

In the case of \(f(1)\), evaluation is finding out what the function \(f(x)\) equals when \(x = 1\). Steps involved in evaluating a function include:

• First, substituting the variable with the given value.
• Next, simplify the expression, just as we discussed.
• Finally, perform the necessary arithmetic to compute the result, which was calculated as \(6\) for our function.

This process confirms the behavior of the function at a specific input, which is essential for understanding how functions work. It applies to many areas in mathematics where interpreting the output of a function based on a given input is crucial.