Problem 51 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \( f(x) = \left\\{ \begin{array}{ll} 3x - 1, & \mbox{ \) x < -1 \(} \\ 4, & \mbox{ \) -1 \le x \le 1 \(} \\ x^2, & \mbox{ \) x > 1 \(} \end{array} \right.\) (a) \(f(-2)\) (b) \(f(-\frac{1}{2})\) (c) \(f(3)\)

Step-by-Step Solution

Answer

\n\(f(-2) = -7\)\n\(f(-\frac{1}{2}) = 4\)\n\(f(3) = 9\)

Step 1: Evaluate \(f(-2)\)

First check which condition the value \(x=-2\) satisfies. In the given function, when \(x=-2\), this fits into the condition \(x < -1\). Therefore for \(f(-2)\), apply \(3x - 1\). This gives us a value of \(3(-2) - 1 = -6 - 1 = -7\).

Step 2: Evaluate \(f(-\frac{1}{2})\)

Next, determine where \(x=-\frac{1}{2}\) fits in. This falls into the condition \(-1 \le x \le 1\). Our function specifies that when \(x\) is in this range, the value is 4, so \(f(-\frac{1}{2}) = 4\).

Step 3: Evaluate \(f(3)\)

Lastly, for the value \(x=3\), this matches with the condition \(x > 1\). Therefore use the function \(x^2\). So, when evaluated, \(f(3) = 3^2 = 9\).

Key Concepts

Function Evaluation

To evaluate a piecewise function, you first need to understand which part of the function applies to the specific value of \( x \) you're working with. This involves looking at the conditions provided in the piecewise function. Each condition covers a different range of \( x \), and the function you use depends on where your \( x \) value lies in these ranges.

• If the value of \( x \) is less than -1, use the expression \( 3x - 1 \).
• If \( x \) is between -1 and 1 inclusive, use the constant value 4.
• For \( x \) greater than 1, employ the quadratic part \( x^2 \).

In our example, we evaluated the function at three different values of \( x \): -2, -\( \frac{1}{2} \), and 3, each fitting a different piece of the function. Always double-check that your \( x \) value fits into the range specified before applying the corresponding expression to avoid errors.

Inequalities

Inequalities are mathematical expressions involving a less than (<), greater than (>), less than or equal to (\( \le \)), or greater than or equal to (\( \ge \)) relation between two values. In piecewise functions, inequalities are used to define which function to use for particular ranges of \( x \).
These inequalities can often be found immediately after each piece of the function's definition. For example:

• \( x < -1 \) implies using the formula \( 3x - 1 \).
• \( -1 \le x \le 1 \) means you simply use 4.
• \( x > 1 \) requires using \( x^2 \).

Understanding these inequalities is crucial. They help you categorize the value of \( x \) you're focusing on, ensuring you use the right piece of the function. Practicing with these can greatly help you feel more comfortable when evaluating functions involving inequalities.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically expressed in the form \( ax^2 + bx + c \). In our piecewise function, the quadratic part is simply \( x^2 \).
This means when choosing where \( x > 1 \), the function behaves according to the rules of a quadratic function. Quadratic functions are known for their characteristic parabola shape when graphed, either opening upwards or downwards, depending on the sign of the coefficient of \( x^2 \).
In the exercise, for \( x = 3 \), using \( x^2 \), we calculated the function value as \( 9 \). This demonstrates the simplicity of calculating quadratic functions - square the \( x \) value. Even though quadratic functions can be more complex with additional linear terms, in this simplified case, they're still just about squaring.

Linear Functions

Linear functions are the most straightforward types of functions characterized by a constant rate of change, typically resembling the format \( ax + b \), where \( a \) and \( b \) are constants. This provides a straight-line graph.
In our piecewise example, for \( x < -1 \), we use the linear function \( 3x - 1 \). This means, whatever \( x \) might be in that range, you will input it, multiply it by 3, and subtract 1 to get your result.
For instance, with \( x = -2 \), using the expression \( 3x - 1 \), we calculated the result as \( -7 \). This linear function aspect of the piecewise function is straightforward and provides a good foundation for understanding how changes in \( x \) affect the output linearly. Linear functions are essential building blocks and understanding them is crucial for grasping more complex function types.