Problem 34 - 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. $$ f(-2) $$

Step-by-Step Solution

Answer

\(f(-2) = -4.5\)

Step 1: Substitute the value into the function

First, substitute \(-2\) into the function \(f(x)\) to evaluate \(f(-2)\). The function is \(f(x) = \frac{x^2 + 5}{x}\). So substitute \(x = -2\) into the expression.

Step 2: Calculate numerator

Calculate the numerator by plugging in the value \(-2\) in place of \(x\) in the expression \(x^2 + 5\).\((-2)^2 + 5 = 4 + 5 = 9\).

Step 3: Calculate denominator

Calculate the denominator by plugging in the value \(-2\) in place of \(x\) in the expression \(x\), which is simply \(-2\).

Step 4: Perform the division

Divide the numerator \(9\) by the denominator \(-2\). \(\frac{9}{-2} = -4.5\).

Key Concepts

Substitution

Substitution is the first critical step when evaluating a function at a specific point. It involves replacing the variable within the function with the given value. In our exercise, the function is \(f(x) = \frac{x^2 + 5}{x}\) and we need to find \(f(-2)\). To achieve this, we substitute \(-2\) in place of \(x\) in the function. This will help us transform the function's general expression into one that depends solely on numbers, making it easier to resolve the remaining calculation steps.

• Identify the variable in the function.
• Replace every instance of the variable with the substitution value (here, \(-2\)).
• This substitution shifts our focus to evaluating a specific expression rather than a general function.

Substitution makes it possible to handle abstract function expressions more concretely.

Numerator Calculation

Once we have substituted the variable with the given value, the next step is to compute the numerator of our expression. In this case, the original numerator is \(x^2 + 5\). After substitution, this becomes \((-2)^2 + 5\). Here’s how we calculate it:

• First, compute \((-2)^2\). Squaring a negative number results in a positive number, so \((-2)^2 = 4\).
• Next, add 5 to 4, which gives us \(9\).

Evaluating the numerator is crucial because this value will be used in the division process to derive the function's final value at \(x = -2\). Careful calculation at this step ensures accuracy in the subsequent steps.

Denominator Calculation

After determining the numerator, our next focus shifts to the denominator, originally simply \(x\) in our function. Substituting \(-2\) for \(x\) results in a very straightforward calculation of the denominator component.

• The denominator is simply \(-2\) as there is no complex expression here. We directly substitute and retain \(-2\) because our denominator is only the variable \(x\).

Calculating the denominator correctly is essential because it is used to divide the numerator, and any mistake here will skew the entire operation. It’s a simple step but pivotal for precise function evaluation.

Division Operation

With both the numerator and denominator calculated, the final step is the division operation. This involves dividing the computed numerator by the computed denominator. It’s the step where all previous work comes together. Here, we have:

• Numerator = 9
• Denominator = \(-2\)

To find \(f(-2)\), perform the division \( \frac{9}{-2} \). This results in \(-4.5\). Division decides how these calculated parts fit together, giving us the final function value at the particular point. Ensure that you handle signs properly: dividing by a negative flips the sign of your result.