If f(x)=x^2, what is the domain and range of y=0.5f(3(x-4))-1?
The domain and range of a polynomial are both (-∞∞).
Evaluate
y=0.5 f(3(x-4)) -1
If y is still a polynomial, then ...
To determine the domain and range of the function y = 0.5f(3(x-4))-1, let's break down the steps:
Step 1: Determine the domain of f(x) = x^2.
The domain of a function is the set of all possible values that x can take. In this case, since f(x) is a polynomial function, the domain is all real numbers.
Step 2: Analyze the function y = 0.5f(3(x-4))-1.
Since this function is obtained by applying several transformations to f(x), let's break it down step by step:
a) 3(x-4): This transformation involves subtracting 4 from x and then multiplying the result by 3. This shift the graph of f(x) horizontally and stretches it vertically.
b) f[3(x-4)]: We apply the function f(x) to the result of the previous step. This means every x-value will be squared.
c) 0.5f[3(x-4)]: Multiply the result in b) by 0.5, which compresses the vertical scale by a factor of 0.5.
d) 0.5f[3(x-4)] - 1: Subtract 1 from the result in c), shifting the graph downward by 1 unit.
Now that we understand the transformations, let's consider the domain and range:
Domain:
The domain is determined by the innermost operation. In this case, it is the expression 3(x-4). As we determined earlier in Step 1, the domain of f(x) is all real numbers, so we don't need to worry about any restrictions. Therefore, the domain of y = 0.5f(3(x-4))-1 is also all real numbers.
Range:
Given that we are starting with f(x) = x^2, the range of this function will always be non-negative real numbers (or zero). However, due to the scalings and transformations applied to f(x), the range of y = 0.5f(3(x-4))-1 will also include negative real numbers. Therefore, the range of this function is all real numbers greater than or equal to -1.
In summary, the domain of y = 0.5f(3(x-4))-1 is all real numbers, and the range is all real numbers greater than or equal to -1.