we have just started learning about functions but i don't entirely understand and if anyone could help on these questions i would really appreciate it!
1)if you have the graph f(x)=^3,with domain x is a real number (cant draw it on here!), then what would the range be? would it be f(x) is a real number?
also, if you had the same graph but domain x>2, would the range be f(x)>8?
2) how do you find the range for (x-2)^2 + 3 where x is a real number?
i'm not really certain about this topic so pleaseeee help if you can thankyou!
1) First one is a cubic, and you should know that the domain and range of any odd-degree polynomial (with no constraints) is all real numbers. So yeah, f(x) is a real number.
You're also right for f(x)>8. As x increases (from 2), f(x)=x^3 increases, so there's no way to get under 2^3. Also, as x tends to infinity, so does x^3.
2) There are two ways to interpret the quadratic:
Vertex form: A quadratic, when put in the form a(x-h)^2+k, has a vertex at (h,k). If a is positive, it opens up, and if a is negative, it opens down.
For (x-2)^2 + 3, a=1, h=2, and k=3. So this quadratic is a parabola that opens up with a vertex at (2,3). Recall that the vertex is always the minimum/maximum of a quadratic.
Transformations: f(x)=x^2 is a function with a range of x≥0 (so the minimum is at 0). f(x)+k is a function that is moved up by 'k' units. Therefore, the minimum is also moved up by 'k' units.