Aussie Maths & Science Teachers: Save your time with SmarterEd
Let `f:R→R, \ f(x)=cos(ax)`, where `a in R text(\{0})`, be a function with the property
`f(x)=f(x+h),` for all `h in Z`
Let `g:D rarr R, \ g(x)=log_(2)(f(x))` be a function where the range of `g` is `[-1,0]`.
A possible interval for `D` is
`B`
`f(x)=cos(alpha x)=f(x+h)=cos(a(x+h))`
`=>a=2pi`
`g(x)=log_(2)(f(x))=log_(2)(f(x+h))=log_(2)(cos(a(x+h)))`
`-1leqlog_(2)(cos(2pi x))leq0`
`(1)/(2)leq cos(2pi x)leq1`
`text(Sketch) \ y=cos(2pi x)\ \ text(by CAS.)`
`text(By inspection of graph,)\ \ (1)/(2)leq cos(2pi x)leq1\ \text{for}\ x in [1,(7)/(6)]`
`=>B`
If `f(x - 1) = x^2 - 2x + 3`, then `f(x)` is equal to
`B`
`text(Let)\ g(x) = f(x – 1)`
`text(Shift)\ g(x)\ text(left 1):`
| `g(x + 1)` | `= (x + 1)^2 – 2(x + 1) + 3` |
| `= x^2 + 2` |
`=> B`