Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Use graphing to determine the domain and range of \(y=f(x)\) and of \(y=|f(x)|\). $$f(x)=(x+1)^{2}-2$$

Step-by-Step Solution

Answer

Domain for both: \((- fty, fty)\); range of \(f(x)\): \([-2, fty)\); range of \(|f(x)|\): \([0, fty)\).

Step 1: Analyze the Original Function

The function given is \( f(x) = (x+1)^2 - 2 \). This is a transformation of the basic quadratic function \( y = x^2 \), which opens upwards with vertex at \( (-1, -2) \). Analyzing its graph shows that it is a parabola that opens upwards, meaning it has no restrictions horizontally, making the domain all real numbers.

Step 2: Determine the Domain of \( f(x) \)

Since the quadratic function \( f(x) = (x+1)^2 - 2 \) is defined for all real values of \( x \), the domain is all real numbers. In interval notation, this is expressed as \((-fty, fty)\).

Step 3: Determine the Range of \( f(x) \)

The vertex \( (-1, -2) \) is the lowest point on the graph of the parabola \( f(x) = (x+1)^2 - 2 \). As the parabola opens upwards, the range starts from \(-2\) and goes to infinity. Therefore, the range of \( f(x) \) is \([-2, fty)\).

Step 4: Analyze the Absolute Value Function \( y = |f(x)| \)

For the absolute value function \( y = |f(x)| \), where \( f(x) = (x+1)^2 - 2 \), all output values of \( f(x) \) are transformed to be non-negative. Values of \( f(x) \) that are already positive or zero remain unchanged, but negative values are reflected across the x-axis to be positive.

Step 5: Determine the Domain of \( y = |f(x)| \)

The domain of the absolute value function \( |f(x)| \) is the same as that of \( f(x) \), as the absolute value transformation does not affect the x-values. Hence, the domain is \((-fty, fty)\).

Step 6: Determine the Range of \( y = |f(x)| \)

Since all values of \( f(x) \) that are below zero are reflected to positive values, the lowest value \( y = |f(x)| \) can have is 0. Consequently, the range of \( y = |f(x)| \) is \([0, fty)\).

Key Concepts

Absolute Value Function

The absolute value function, often denoted as \(|x|\), transforms any number into its non-negative value. This means that while positive numbers and zero remain unchanged, negative numbers become positive. For example, \(|3| = 3\) and \(|-2| = 2\). This concept is crucial in understanding how graphs transform when using absolute value.

When we apply the absolute value to a function, say \(y = |f(x)|\), it modifies the output of the original function \(f(x)\) such that any negative result is reflected to become positive. As a result, the graph appears as a "v" shape or a mirrored reflection of the negative parts onto the positive side.

In terms of determining the range of \(|f(x)|\), the lowest value for \(y\) will be 0. This is because the absolute value ensures no negative outputs. For the given function \(f(x) = (x+1)^2 - 2\), the range of the original function was \([-2, \infty)\), but the range of \(|f(x)|\) is adjusted to \([0, \infty)\). This change illustrates how absolute value impacts the set of possible outputs.

Quadratic Function

A quadratic function is a type of polynomial function represented by the equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Its graph is a parabola, which can open upwards or downwards depending on the sign of \(a\).

In our function \(f(x) = (x+1)^2 - 2\), it is a transformation of \(y = x^2\), a basic upward-opening parabola. The vertex form \((x-h)^2 + k\) helps identify transformations easily. Here, \(h = -1\) shifts the parabola to the left, and \(k = -2\) moves it down on the y-axis. Consequently, the vertex, or the lowest point, is \((-1, -2)\).

Understanding this helps us determine the domain and range. The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range, however, depends on whether the parabola opens upwards or downwards. For our function, which opens upwards, the range starts at the vertex's y-coordinate and extends to positive infinity, or \([-2, \infty)\).

Interval Notation

Interval notation is a mathematical notation used to represent sets of numbers along a number line. It is particularly useful for describing domains and ranges of functions.

There are two types of brackets used in interval notation:

• Round brackets, \((\) or \())\), which indicate that endpoints are not included (open interval).
• Square brackets, \([\) or \(]\), which indicate that endpoints are included (closed interval).

For example, the interval \((a, b)\) represents all numbers greater than \(a\) and less than \(b\), but not including \(a\) and \(b\) themselves. An interval like \([a, b]\) includes both endpoints \(a\) and \(b\).

In terms of function analysis, knowing how to write intervals precisely is crucial. For instance, the domain of \(f(x) = (x+1)^2 - 2\) is all real numbers, represented as \((-\infty, \infty)\). The range \([-2, \infty)\) indicates that the parabola reaches its lowest y-value at \(-2\) and extends upwards indefinitely. Interval notation provides a clear and concise way to capture these aspects of a function.