👉 A particular function, \mathrm{gd}(x)=2\tan^{-1}e^x-\frac\pi2.
Alright class, settle down now, let’s take a look! You’ve stumbled upon a notation that might seem a bit strange – “gd(x)”. It's a shorthand we use in math to represent a specific function. The dictionary you consulted is giving you the key: gd(x) = 2tan⁻¹eˣ - π/2 Let’s break that down, nice and slow. gd(x) simply tells us we're talking about this particular function . Think of it like a label! tan⁻¹ is short for “tangent inverse,” or sometimes written as arctan. It's the inverse operation of the tangent function - if you know the tangent of an angle, this will give you the angle itself. (Remember, tan θ = opposite/adjacent!) eˣ means "e to the power of x," where 'e' is a special number (approximately 2.718) – it’s a constant! π/2 is just pi divided by two. Pi (π) is approximately 3.14159, so π/2 is about 1.5708. So, the function gd(x) calculates something involving the tangent of eˣ, then subtracts π/2 from that result. It’s a specific transformation! Think of it like this: Imagine you're building with blocks. ‘gd(x)’ is telling you to use a certain recipe - take 'eˣ', find its tangent, and then adjust the answer by subtracting a fixed amount (π/2). Do you have any questions about what that means or how we might work with this function? Let's discuss!
