The equation has a real solution. Choose the official best answer describing the attainable range of k.

01 / Standard#Your turn

9^{-x^2}-4\cdot3^{-x^2}=k

Official paper · jm01-2021 · I.7 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Set u=3^(−x²) and find its exact range.

Hint 2

On 0<u≤1, the expression u²−4u decreases.

Worked solution

The exponent is nonpositive, and a positive exponential never reaches zero.
$u=3^{-x^2}\in(0,1]$
Compare endpoints and the unattained limiting value.
$k=u^2-4u=(u-2)^2-4\in[-3,0)$
Every u in (0,1] is attainable by a real x.
$x=\pm\sqrt{-\log_3u}$

C: −3≤k<0 is the exact range.

Checks and common pitfalls: Do not include k=0: it is a limiting value, not an attained one.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The exponent is nonpositive, and a positive exponential never reaches zero.
$u=3^{-x^2}\in(0,1]$
Compare endpoints and the unattained limiting value.
$k=u^2-4u=(u-2)^2-4\in[-3,0)$
Every u in (0,1] is attainable by a real x.
$x=\pm\sqrt{-\log_3u}$

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

2021 JM01