Problem 50 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \( f(x) = \left\\{ \begin{array}{ll} x^2 + 2, & \mbox{ \) x \le 1 \(} \\ 2x^2 + 2, & \mbox{ \) x > 1 \(} \end{array} \right.\) (a) \(f(-2)\) (b) \(f(1)\) (c) \(f(2)\)

Step-by-Step Solution

Answer

The values of \(f(-2)\), \(f(1)\), and \(f(2)\) are 6, 3, and 10 respectively.

Step 1: Evaluate f(-2)

First, check which piece of the function to use. Because -2 is less than 1, we use the first equation. Thus, \(f(-2) =(-2)^2 + 2 = 4 + 2 = 6\).

Step 2: Evaluate f(1)

Since 1 is equal to 1, we use the first equation again. So, \(f(1) = (1)^2 + 2 = 1 + 2 = 3\).

Step 3: Evaluate f(2)

For this one, 2 is greater than 1, so we use the second equation. Hence, \(f(2) = 2*(2)^2 + 2 = 2*4 + 2 = 10\).

Key Concepts

Function Evaluation

Function evaluation is about finding the value of a function corresponding to a specific input. In mathematics, a function can be viewed as a machine that takes an input, performs operations, and delivers an output. Here, the function is given as a rule or formula that we apply to the supplied values to get results. Understanding function evaluation is essential in mathematics as it helps in observing how changes in input affect outputs.
To evaluate a function, follow these steps:

• Identify the input value you wish to evaluate.
• Determine which rule or formula of the function you need to apply (especially important in piecewise functions).
• Substitute the input value into the correct part of the formula
• Simplify to get the result.

In our example, we're given a piecewise function, and we need to plug in values like -2, 1, or 2 and simplify the expression according to the rules given in each piece of the function.

Quadratic Function

A quadratic function is a type of polynomial function that can be expressed in the form: \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. Quadratic functions create a symmetrical curve known as a parabola when graphed on a coordinate plane. These functions are fundamental in algebra and appear frequently in various mathematical exercises.
A closer look at our initial function reveals a quadratic nature in both pieces:

• For \( x \le 1 \), the expression \( x^2 + 2 \) is quadratic as it contains the \( x^2 \) term.
• For \( x > 1 \), the expression \( 2x^2 + 2 \) includes the same \( x^2 \) term but is scaled by a factor with its coefficients adjusted.

The signature \( x^2 \) term is what makes them both quadratic. Evaluating quadratic expressions involves common steps:

• Identifying the quadratic components from the equation.
• Substituting specific values of \( x \) as given.
• Performing calculations following the order of operations to find the result.

Piecewise-Defined Functions

Piecewise-defined functions are special mathematical functions defined by multiple sub-functions, each over a specific interval of the domain. These types of functions can appear complex at first but become easier to understand with practice. Here's what to keep in mind when dealing with piecewise functions:

• Each sub-function relates a different input domain to a particular output.
• Identify which part of the function should be applied based on the particular value of \( x \).
• It's crucial to pay attention to the inequality symbols. They dictate which sub-function you should use.

For the function provided in the exercise, the domain splits at \( x = 1 \). The function changes its rule when the input crosses this boundary:

• When \( x \le 1 \), you use the expression \( x^2 + 2 \).
• For values where \( x > 1 \), the expression \( 2x^2 + 2 \) is applied.

Practicing with these functions will help you become proficient at pinpointing the correct rule to apply for any given input.