Video Transcript
The range of the function 𝑓 of 𝜃 equals 𝑎 cos of three 𝜃 is the closed interval from negative five over four to five over four. Find the value of 𝑎 where 𝑎 is greater than zero.
The range of a function is all of its possible outputs. And so, since the range of our function is the closed interval from negative five over four to five over four, we can say that 𝑎 cos of three 𝜃 must always be greater than or equal to negative five over four and less than or equal to five over four. But how does this help us? Well, let’s just consider cos of 𝜃. The range of the function 𝑓 of 𝜃 is equal to cos of 𝜃 is the closed interval from negative one to one. So, we can say that cos of 𝜃 must always be greater than or equal to negative one and less than or equal to one.
What effect does multiplying 𝜃 by three have on the range of our function? Well, if we think about mapping the function cos 𝜃 onto cos of three 𝜃, we know that it’s a horizontal stretch by a scale factor of one-third. This doesn’t change the range; it doesn’t change the output. It will still produce values greater than or equal to negative one and less than or equal to one.
Our function, of course, is 𝑎 cos of three 𝜃. So, what’s the transformation here? To map cos of three 𝜃 onto 𝑎 times cos of three 𝜃, we have a vertical stretch by a scale factor of 𝑎. This means the minimum output will now be negative 𝑎, that’s negative one times 𝑎, and the maximum output will be 𝑎, that’s one times 𝑎. We can now compare our original inequality to our new inequality. We see we have 𝑎 cos of three 𝜃 in both inequalities. We then have negative five-quarters and negative 𝑎 and five-quarters and 𝑎. This means we can deduce that 𝑎 must be equal to five over four. This is indeed greater than zero, as required. And so, 𝑎 must be equal to five over four.
