Daily Math · 2026-06-30

Question 1

Let $f(x) = 3\log_2 x$. Find $f^{-1}(27)$.

Accepted Answer

$\bullet$ Invert $y = 3\log_2 x$:

$$\log_2 x = 	frac{y}{3} \;\Rightarrow\; x = 2^{y/3}$$

$\bullet$ So $f^{-1}(y) = 2^{y/3}$.

$\bullet$ Evaluate at $y = 27$:

$$f^{-1}(27) = 2^{27/3} = 2^9 = 512$$

$\bullet$ Check: $f(512) = 3\log_2 512 = 3 \cdot 9 = 27$.

Answer

$9$

Answer

$128$

Answer

$2^{27}$

Answer

$512$

Answer

$\dfrac{1}{512}$

Answer

$256$

Question 2

How many solutions does $\cos 2x = -\dfrac{1}{2}$ have on $[0, 2\pi)$?

Accepted Answer

$\cos u = -	frac{1}{2}$ on $[0, 4\pi)$ at $u = 	frac{2\pi}{3}, 	frac{4\pi}{3}, 	frac{8\pi}{3}, 	frac{10\pi}{3}$. Dividing by $2$: $x = 	frac{\pi}{3}, 	frac{2\pi}{3}, 	frac{4\pi}{3}, 	frac{5\pi}{3}$. Four solutions.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$6$

Question 3

$\displaystyle\int_0^{\infty} e^{-x^2} \,dx$ equals:

Accepted Answer

$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$. By even symmetry, $\int_0^{\infty} e^{-x^2}\,dx = \dfrac{\sqrt{\pi}}{2}$.

Answer

$	frac{\sqrt{\pi}}{2}$

Answer

$\sqrt{\pi}$

Answer

$	frac{\pi}{2}$

Answer

$1$

Problems & worked solutions

Problem 1 — Logs

Problem 2 — Trigonometric Equations

Problem 3 — Calculus