Question

A function, h(x), is defined as shown.

h(x) = StartLayout Enlarged left-brace 1st row 1st column one-fourth x minus 4, 2nd column x less-than-or-equal-to 0 2nd row 1st column one-third x minus 3, 2nd column 0 less-than x less-than-or-equal-to 3, 3rd row 1st column one-half x minus 2, 2nd column x greater-than-or-equal-to 4

Which graph represents h(x)?

On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 4, 0) and goes up to a closed circle at (0, 1). The second line starts at (0, negative 3) and goes up to a closed circle at (3, negative 2). The third line has a closed circle at (4, negative 2) and goes up to (6, negative 1).

On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 4, negative 5) and goes up to a closed circle at (0, negative 4). The second line starts at (0, negative 3) and goes up to a closed circle at (3, negative 2). The third line has a closed circle at (4, 0) and goes up to (6, 1).

On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 4, 3) and goes up to a closed circle at (0, 4). The second line starts at (0, 3) and goes up to a closed circle at (3, 4). The third line has a closed circle at (4, 4) and goes up to (6, 5).

On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 4, 0) and goes up to a closed circle at (0, 1). The second line starts at (0, negative 1) and goes up to a closed circle at (3, 0). The third line has a closed circle at (4, negative 2) and goes up to (6, negative 1).

Answer 1

h(x) = { (1/4)x − 4 for x ≤ 0; (1/3)x − 3 for 0 < x ≤ 3; (1/2)x − 2 for x ≥ 4 }.

Evaluate endpoints:
- For x = −4: (1/4)(−4) − 4 = −5 → first segment starts at (−4, −5).
- For x = 0 (first piece, closed): (1/4)(0) − 4 = −4 → first segment ends at closed (0, −4).
- For x → 0+ (second piece, open): (1/3)(0) − 3 = −3 → second segment starts at open (0, −3).
- For x = 3: (1/3)(3) − 3 = −2 → second segment ends at closed (3, −2).
- For x = 4: (1/2)(4) − 2 = 0 → third segment starts at closed (4, 0).
- For x = 6: (1/2)(6) − 2 = 1 → third segment goes to (6, 1).

These match the second graph (the one with (−4, −5) → closed (0, −4); open (0, −3) → closed (3, −2); closed (4, 0) → (6, 1)).