Problem 62 - Algebra and Trigonometry

Question

If \(f(x)=\frac{x^{2}+1}{2 x+5},\) find \(f(-2) .\) What is the corresponding point on the graph of \(f ?\)

Step-by-Step Solution

Answer

The function value is 5, and the corresponding point is (-2, 5).

Step 1 - Identify the Function

The given function is \( f(x) = \frac{x^2 + 1}{2x + 5} \). To find the value of the function at a specific point, substitute that point into the function.

Step 2 - Substitute the Value of x

We need to find \( f(-2) \). Substitute \( x = -2 \) into the function: \( f(-2) = \frac{(-2)^2 + 1}{2(-2) + 5} \).

Step 3 - Simplify the Numerator

Simplify the expression in the numerator: \( (-2)^2 + 1 = 4 + 1 = 5 \).

Step 4 - Simplify the Denominator

Simplify the expression in the denominator: \( 2(-2) + 5 = -4 + 5 = 1 \).

Step 5 - Calculate the Value of the Function

Now, divide the simplified numerator by the simplified denominator: \( f(-2) = \frac{5}{1} = 5 \).

Step 6 - Identify the Corresponding Point on the Graph

The corresponding point on the graph is \((-2, 5)\).

Key Concepts

Rational Functions

A rational function is a type of function represented by the quotient of two polynomials. The general form is: \[ f(x) = \frac{P(x)}{Q(x)} \]
where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero. In our exercise, the given rational function is: \[ f(x) = \frac{x^2 + 1}{2x + 5} \] Understanding rational functions is key to seeing how the values of these functions change as different values are substituted into them.

Substitution

Substitution is the process of replacing a variable in a function with a specific value. This helps determine the output of the function for that particular input.
Let's look at this through the given problem step by step:

• Start with the function \( f(x) = \frac{x^2 + 1}{2x + 5} \)
• To find \( f(-2) \), substitute \( x = -2 \) into the function.
• This gives us: \[ f(-2) = \frac{(-2)^2 + 1}{2(-2) + 5} \]

Substitution is an essential method in calculus and algebra to find specific values of functions.

Simplifying Expressions

Simplifying expressions is the process of rewriting a complicated expression in a simpler form.
For our given problem, we simplify both the numerator and the denominator:

First, simplify the numerator: \( (-2)^2 + 1 = 4 + 1 = 5 \)

Next, simplify the denominator: \( 2(-2) + 5 = -4 + 5 = 1 \)

By simplifying these parts, we make it easier to find the final value of the function:
\[ f(-2) = \frac{5}{1} = 5 \] Simplification is crucial for obtaining results that are easier to understand and use.

Graphing Points

Graphing points is a key aspect of visualizing the behavior of functions.
Each point on a graph corresponds to an input-output pair of the function. For instance, in our exercise, after finding that \( f(-2) = 5 \), we identify the corresponding point on the graph, which is \((-2, 5)\).
To graph this:

• Locate the \(x\)-coordinate \(-2\) on the horizontal axis.
• Locate the \(y\)-coordinate \(5\) on the vertical axis.
• Place a dot where these two coordinates meet.

Graphing points helps understand the overall shape and behavior of the function.
Pivoting points on the graph can also reveal important features like intercepts, maxima, and minima.