Question

1. \\( f(x) = \

$$\begin{cases} x^2 + 2, & \\text{if } x < 0 \\\\ x, & \\text{if } x \\ge 0 \\end{cases}$$

\\)

1. \\( g(x) = \

$$\begin{cases} x + 2, & \\text{if } x \\le 0 \\\\ x - 2, & \\text{if } x > 0 \\end{cases}$$

\\)

1. \\( h(x) = \

$$\begin{cases} x^3, & \\text{if } x \\le 0 \\\\ \\sqrt{x}, & \\text{if } x > 0 \\end{cases}$$

\\)

1. \\( k(x) = \

$$\begin{cases} x - 1, & \\text{if } x < 0 \\\\ x^2 + 1, & \\text{if } x > 0 \\end{cases}$$

\\)

Explanation:

Analyze the given piecewise function equations

Using the Piecewise Functions and Identifying Piecewise Functions knowledge points, we examine the definitions of the four functions:

1. \(f(x) =

$$\begin{cases} x^2 + 2, & \text{if } x < 0 \\ x, & \text{if } x \ge 0 \end{cases}$$

\)

1. \(g(x) =

$$\begin{cases} x + 2, & \text{if } x \le 0 \\ x - 2, & \text{if } x > 0 \end{cases}$$

\)

1. \(h(x) =

$$\begin{cases} x^3, & \text{if } x \le 0 \\ \sqrt{x}, & \text{if } x > 0 \end{cases}$$

\)

1. \(k(x) =

$$\begin{cases} x - 1, & \text{if } x < 0 \\ x^2 + 1, & \text{if } x > 0 \end{cases}$$

\)

Match function 1 to its graph

Using the Piecewise Functions knowledge point:

• For \(x < 0\), the graph is \(y = x^2 + 2\), which is a parabola opening upwards with vertex at \((0, 2)\). Since \(x < 0\), there is an open circle at \((0, 2)\).
• For \(x \ge 0\), the graph is \(y = x\), which is a straight line starting at \((0, 0)\) with a solid circle and going up to the right with slope 1.
• This matches graph d.

Match function 2 to its graph

Using the Piecewise Functions knowledge point:

• For \(x \le 0\), the graph is \(y = x + 2\), which is a line with slope 1 and y-intercept \((0, 2)\) (solid circle).
• For \(x > 0\), the graph is \(y = x - 2\), which is a line with slope 1 starting with an open circle at \((0, -2)\).
• The top graph shown in the image (partially cropped at the top) shows a line with slope \(0.5\) ending at an open circle at \((0, -1)\). This corresponds to \(y = \frac{1}{2}x - 1\) for \(x < 0\), which is not listed among the options. However, we only need to match the visible lettered graphs.

Match function 3 to its graph

Using the Square Root Function and Piecewise Functions knowledge points:

• For \(x \le 0\), the graph is \(y = x^3\), which passes through \((0, 0)\), \((-1, -1)\), and \((-2, -8)\).
• For \(x > 0\), the graph is \(y = \sqrt{x}\), which starts at \((0, 0)\) and passes through \((1, 1)\), \((4, 2)\).
• This matches graph c.

Answer:

Question 1

\(f(x)\) matches graph d.

Question 3

\(h(x)\) matches graph c.