Determine the range of a quadratic g when the composite has range [0,∞).

01 / Standard#Your turn

f(t)=\begin{cases}\sqrt{|t|},&|t|\ge4,\\t,&|t|<4.\end{cases}

Official paper · jm01-2026 · I.15 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A quadratic range is a single closed half-line.

Hint 2

f(t)=0 only at t=0; negative t between −4 and 0 give negative outputs.

Worked solution

The range of g must include 0 but cannot include any t in (−4,0). A downward half-line through 0 is therefore impossible.
The only upward half-line satisfying these conditions starts at 0; its image under f is the desired range.
$g(\mathbb R)=[0,\infty),\quad f([0,\infty))=[0,4)\cup[2,\infty)=[0,\infty)$

C: [0,∞).

Checks and common pitfalls: A disconnected set cannot be the range of a real quadratic.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The range of g must include 0 but cannot include any t in (−4,0). A downward half-line through 0 is therefore impossible.
The only upward half-line satisfying these conditions starts at 0; its image under f is the desired range.
$g(\mathbb R)=[0,\infty),\quad f([0,\infty))=[0,4)\cup[2,\infty)=[0,\infty)$

Think first. Reveal a hint when the class is ready.

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