Question

(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. $$f(x)=\left\\{\begin{array}{ll}\frac{1}{x} & \text { if } x<0 \\\\\sqrt[3]{x} & \text { if } x \geq 0\end{array}\right.$$

Short answer

Domain: \( (-\infty, \infty) \), Intercepts: (0,0), Range: \( (-\infty, \infty) \).

Step-by-step solution

Find the Domain

The domain of the function is the set of all possible input values (x-values). For the given piecewise function: When \( x < 0 \), the function is \( f(x) = \frac{1}{x} \). For this part, the domain is all negative numbers since dividing by zero is undefined. When \( x \geq 0 \), the function is \( f(x) = \sqrt[3]{x} \). The cube root function is defined for all real numbers, including zero. Thus, the overall domain of the function is all real numbers: \( (-\infty, \infty) \).

Locate Intercepts

An intercept is where the graph crosses the axis: * **y-intercept**: \( x = 0 \) For \( x = 0 \), the function is \( f(0) = \sqrt[3]{0} = 0 \). So the y-intercept is at \( (0, 0) \). * **x-intercept**: \( y = 0 \) For \( y = 0 \), we need to solve: - \( \frac{1}{x} = 0 \) has no solution since \( \frac{1}{x} \) cannot be zero. - \( \sqrt[3]{x} = 0 \) implies \( x = 0 \).Therefore, the x-intercept is also at \( (0, 0) \).

Graph the Function

Graph the piecewise function: - For \( x < 0 \), graph \( f(x) = \frac{1}{x} \). This is a hyperbola located in the second and third quadrants. - For \( x \geq 0 \), graph \( f(x) = \sqrt[3]{x} \). This is a curve that passes through the origin and increases gradually in the first quadrant.

Find the Range Based on the Graph

The range of the function is the set of all possible output values (y-values). Based on the graph: - For \( x < 0 \), \( f(x) = \frac{1}{x} \) can take any negative or positive value except zero. - For \( x \geq 0 \), \( f(x) = \sqrt[3]{x} \) can take any non-negative value, including zero.Thus, the overall range is all real numbers: \( (-\infty, \infty) \).

Key concepts

Domain

In mathematics, the domain of a function is the set of all possible input values (x-values) that the function can accept. For our given piecewise function:
When \( x < 0 \), the function is \( f(x) = \frac{1}{x} \). Here, we can use any negative value for \( x \) since dividing by zero is undefined but all other negative values are allowed.
When \( x \geq 0 \), the function is \( f(x) = \sqrt[3]{x} \). The cube root function is defined for all real numbers, including zero.
So, combining both parts, the domain for our function is all real numbers, which we write as \( (-\infty, \infty) \). This means you can plug any real number into this function, and it will work.

Intercepts

Intercepts are points where a graph crosses either the x-axis or the y-axis:
• **Y-intercept:** This is where the graph crosses the y-axis; it happens when \( x = 0 \). For our function, \( f(0) = \sqrt[3]{0} = 0 \). Thus, the y-intercept is at \( (0, 0) \).
• **X-intercept:** This is where the graph crosses the x-axis; it occurs when \( y = 0 \). To find the x-intercept, we check when the function values equal zero.
For \( f(x) = \frac{1}{x} \), there is no solution because \( \frac{1}{x} \) can never be zero.
For \( f(x) = \sqrt[3]{x} \), setting \( \sqrt[3]{x} = 0 \) implies \( x = 0 \). Thus, the x-intercept is also at \( (0, 0) \). Therefore, the intercepts are important for understanding the behavior of the function at key points.

Graphing Functions

Graphing a piecewise function involves plotting different functions over specified intervals:
• For \( x < 0 \), we graph \( f(x) = \frac{1}{x} \). This forms a hyperbola located in the second and third quadrants since our domain covers negative x-values.
• For \( x \geq 0 \), we graph \( f(x) = \sqrt[3]{x} \). This creates a curve that passes through the origin and increases gradually in the first quadrant.
By plotting both these parts on the same coordinate plane, we get a complete visualization of the piecewise function. This helps in understanding each section's behavior and how they join at \( x = 0 \).

Range

The range of a function is the set of all possible output values (y-values) that the function can produce:
• For \( x < 0 \), \( f(x) = \frac{1}{x} \) can produce any negative or positive value except zero.
• For \( x \geq 0 \), \( f(x) = \sqrt[3]{x} \) produces non-negative values including zero.
By combining both parts, the overall range of our function is all real numbers, written as \( (-\infty, \infty) \). This means the function can output any real number. Understanding the range is crucial for fully grasping the behavior and scope of the function.