What are the domain and range of the function $y = 2 ext{cos}^{-1}(2x) + 2 ext{sin}^{-1}(2x)$? - HSC - SSCE Mathematics Extension 1 - Question 4 - 2024 - Paper 1

Next, we will determine the range of the function. The expressions for the inverse cosine and sine yield outputs that we can analyze:

1. The output of $ext{cos}^{-1}(u)$ where $u ext{ in } [-1, 1]$ is in the interval $[0, ext{π}]$.

2. The output of $ext{sin}^{-1}(u)$ is in the interval [-rac{ ext{π}}{2}, rac{ ext{π}}{2}].

Since the function combines these outputs:

• The contribution from $2 ext{cos}^{-1}(2x)$ gives a range of $[0, 2 ext{π}]$.
• The contribution from $2 ext{sin}^{-1}(2x)$ gives a range of $[- ext{π}, ext{π}]$.

To find the combined range, we add the intervals for each function and analyze the possible values:

The minimum value occurs at $x = -0.5$ yielding:
y_{ ext{min}} = 2( ext{π}) - 2rac{ ext{π}}{2} = 0

The maximum occurs at $x = 0.5$ yielding:
$y_{ ext{max}} = 0 + 2( ext{π}) = 2 ext{π}$

Thus, the overall range is: $[- ext{π}, 2 ext{π}]$