Problem 47 - Algebra A Combined Function

Question

Sketch the graph of each piecewise-defined function. Write the domain and range of each function. $$ f(x)=\left\\{\begin{array}{rll} |x| & \text { if } & x \leq 0 \\ x^{2} & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \([0, \infty)\).

Step 1: Understand the Function Definition

The function is piecewise, meaning it is defined by different expressions based on the value of \(x\). For \(x \leq 0\), \(f(x) = |x|\), and for \(x > 0\), \(f(x) = x^2\).

Step 2: Graph the First Piece: \(f(x) = |x|\) for \(x \leq 0\)

\(|x|\) for \(x \leq 0\) is equivalent to \(-x\) since \(x\) is non-positive in this domain. This part is a line with a slope of -1 from the origin to negative infinity along the \(x\)-axis.

Step 3: Graph the Second Piece: \(f(x) = x^2\) for \(x > 0\)

\(x^2\) is a parabola that opens upwards. This part of the graph starts just to the right of the y-axis (excluding \(x=0\)) and moves upwards into the first quadrant.

Step 4: Determine the Domain of the Function

The domain consists of all values that \(x\) can take. Since \(f(x)\) is defined for \(x \leq 0\) and \(x > 0\), the domain is all real numbers: \((-\infty, \infty)\).

Step 5: Determine the Range of the Function

The range of a function is the set of all possible \(f(x)\) values. As \(x\) goes to negative infinity, \(f(x) = |x|\) reaches infinity, and as \(x\) approaches 0 from the negative side, \(|x|\) approaches 0. For \(x > 0\), \(x^2\) starts just above 0 and goes to infinity. Thus, the range is \([0, \infty)\).

Key Concepts

Absolute Value Function

The absolute value function is a fascinating mathematical concept that represents the distance of a number from zero, regardless of its direction - positive or negative. In simpler terms, the function

• The absolute value of a number \( x \) is written as \( |x| \).
• If the number is positive or zero, its absolute value is the number itself: \( |x| = x \) when \( x \geq 0 \).
• If the number is negative, its absolute value is the positive counterpart: \( |x| = -x \) when \( x < 0 \).

In the context of our exercise, we consider the piece of the function where \( f(x) = |x| \) if \( x \leq 0 \). For these values, \( |x| \) simplifies to \(-x\) because \( x \) is non-positive. The graph of this part of the function will be a straight line descending from the origin with a slope of -1, indicating that as we move left in the number line, the values get more negative.
This graphical representation emphasizes that an absolute value function resembles a 'V' shape when plotted, especially when considering both positive and negative domains together.

Quadratic Function

A quadratic function is one of the most common parabolic functions found in algebra. It is expressed in the standard form of \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a \) is non-zero.Key characteristics of quadratic functions:

• The graph is a parabola, which can either open upwards or downwards depending on the sign of \( a \).
• If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• The vertex of the parabola is its turning point, which can be a maximum or minimum point.

For our piecewise function, the quadratic part \( f(x) = x^2 \) applies when \( x > 0 \). Since \( x^2 \) has a positive coefficient, the parabola opens upwards. It begins right after the y-axis, just as the values of \( x \) start being positive. This upward curve flows into the first quadrant, growing larger as \( x \) increases. Understanding the behavior of this parabolic segment helps us to further comprehend how the function behaves over its domain.

Domain and Range

Domain and range are fundamental concepts in understanding functions and their graphs. The domain of a function is the complete set of possible input values (\( x \) values) that a function can accept.

• The domain encompasses all real numbers that can be plugged into the function without resulting in undefined terms.

In our exercise, the domain of the piecewise function is all real numbers \((-\infty, \infty)\) because the function is defined for both parts \(|x|\) and \(x^2\) over all \(x \leq 0\) and \(x > 0\).

The range, on the other hand, involves all possible output values (\( y \) values) that the function can produce.

• For the absolute portion, as \( x \) values decrease, the values of \( f(x) \) can range from any real number up to just below zero.
• For the quadratic portion, the \( x^2 \) values start just above zero and ascend towards positive infinity.

Thus, putting together outputs from both pieces, the range of the function is \([0, \infty)\), indicating that the function reaches a minimum value of zero (at the point where \( x \) transitions from negative to just above zero) and extends upwards indefinitely. By thoroughly understanding both the domain and range, one can preview how a function behaves and what kind of output it can produce given any input.