Problem 49 - College Algebra

Question

For the following exercises, given each function \(f\) , evaluate \(f(-1), f(0), f(2),\) and \(f(4)\) $$ f(x)=\left\\{\begin{array}{ll}{7 x+3} & {\text { if } x<0} \\ {7 x+6} & {\text { if } x \geq 0}\end{array}\right. $$

Step-by-Step Solution

Answer

\(f(-1) = -4\), \(f(0) = 6\), \(f(2) = 20\), \(f(4) = 34\).

Step 1: Evaluate f(-1)

Since -1 is less than 0, use the piece of the function for \(x < 0\): \(f(x) = 7x + 3\). Substitute \(-1\) into this equation: \(f(-1) = 7(-1) + 3 = -7 + 3 = -4\).

Step 2: Evaluate f(0)

Since 0 is equal to 0, we use the piece of the function for \(x \geq 0\): \(f(x) = 7x + 6\). Substitute \(0\) into this equation: \(f(0) = 7(0) + 6 = 0 + 6 = 6\).

Step 3: Evaluate f(2)

Since 2 is greater than 0, we use the function for \(x \geq 0\): \(f(x) = 7x + 6\). Substitute \(2\) into this equation: \(f(2) = 7(2) + 6 = 14 + 6 = 20\).

Step 4: Evaluate f(4)

Since 4 is greater than 0, we use the function for \(x \geq 0\): \(f(x) = 7x + 6\). Substitute \(4\) into this equation: \(f(4) = 7(4) + 6 = 28 + 6 = 34\).

Key Concepts

Function Evaluation

Function evaluation involves finding the value of a function for specific inputs. In mathematical terms, you take an input value, substitute it into the function, and calculate the result. This is exactly what we did in our exercise example.When evaluating piecewise functions, each part of the function is applied based on specific conditions. For example, when evaluating the given piecewise function for different values of \(x\), we determine which equation to use:

• For \(x = -1\) (less than 0), we use the equation \(f(x) = 7x + 3\).
• For \(x = 0\) (equal to 0), we use the equation \(f(x) = 7x + 6\).
• For \(x = 2\) and \(x = 4\) (both greater than 0), we also use the equation \(f(x) = 7x + 6\).

By assessing which part of the function applies to our input \(x\), we can correctly evaluate the function at a specific point. This is a crucial step in understanding piecewise functions.

Function Domain

The domain of a function is a set of all possible input values for which the function is defined. In simpler terms, it's the range of values we can plug into our function without causing any issues like division by zero or taking the square root of a negative number.In the case of piecewise functions, each piece often has its own domain. For example, in our exercise, the function has two pieces:

• The first part: \(7x + 3\), is used for \(x < 0\).
• The second part: \(7x + 6\), is used for \(x \geq 0\).

Thus, the domain of our piecewise function is all real numbers, since there are pieces defined for every real number. Understanding how each piece corresponds to different parts of the domain helps us evaluate and graph these functions correctly.

Step Functions

Step functions are a specific type of piecewise function that remains constant over intervals but jumps to different values at certain points. Although our exercise focused on linear parts rather than flat steps, this is a good opportunity to understand how these functions operate.In a classic step function, the graph would look like a collection of horizontal segments or 'steps'. They often change values at specific points, creating a 'staircase' effect. Step functions are useful in modeling situations where value changes occur in discrete jumps rather than gradually.In our example, although the function isn't a literal step function, the notion of changing rules at certain values of \(x\) is similar to how step functions switch segments. Knowing how to handle these transitions is key to working with all types of piecewise functions.