Suppose a function f(x) has a domain of (-inf,inf) and range [-11,3]. If we define a new function g(x) by g(x)=f(6x) +1, then what is the range of g(x)? Express your answer in interval notation. Thank you!
Since the domain of f is all x, the range of f(6x) is the same as the range of f(x). The graph is just contracted horizontally by a factor of 6. The range is unaffected.
So, the range of g(x) = f(6x)+1 is just [-11,3]+1 = [-10,4]
Things would have been stickier had the domain been restricted. Then we might not know whether g(x) was even defined for all the values in the domain of f(x).
Suppose a function f(x) is defined on the domain [-8,4]. If we define a new function g(x) by g(x) = f(-2x), then what is the domain of g(x)? Express your answer in interval notation.
To find the range of the function g(x), we need to consider the range of the original function f(x) after the given transformations.
First, let's consider the transformation f(6x). This transformation stretches the graph of f(x) horizontally by a factor of 1/6. Since f(x) has a range of [-11,3], the transformation f(6x) will still have the same vertical range but will be compressed horizontally.
Next, we add 1 to f(6x) to define g(x). This vertical shift does not change the range of f(6x), but it shifts the range upwards by 1.
Therefore, the range of g(x) will be the range of f(x) after being horizontally compressed by a factor of 1/6 and shifted vertically upwards by 1.
The resulting range of g(x) can be expressed in interval notation as:
[-11/6 + 1, 3/6 + 1]
Simplifying, the range of g(x) is:
[-5/6, 1/2]
Therefore, the range of g(x) is (-5/6, 1/2].
To find the range of the function g(x), we first need to determine the range of the original function f(x) and then analyze how the transformation f(6x) + 1 affects the range.
Given that the range of f(x) is [-11, 3], this means that all possible output values of f(x) lie between -11 and 3, inclusive.
Now, let's consider the transformation g(x) = f(6x) + 1. The multiplication by 6 stretches the domain of f(x) by a factor of 1/6, while the addition of 1 shifts the range of f(x) upward by 1 unit.
Since multiplying x by a positive number does not change the direction of the inequality, the domain of g(x) remains unchanged: (-∞, ∞).
As for the range of g(x), we need to consider the effects of the transformation on the range of f(x). By adding 1 to the range of f(x), the range of g(x) shifts upward by 1 unit as well. Therefore, the new range becomes [-11+1, 3+1] = [-10, 4].
Thus, the range of g(x) is [-10, 4] in interval notation.