The range of a function consists of all possible output values (y-values) it can produce based on its input values (x-values).
For the function f(x) = 45x, since 45 is a positive constant, the function will produce all real numbers as x varies over all real numbers. As x approaches positive or negative infinity, f(x) also approaches positive or negative infinity. Therefore, the range of f(x) is:
- Range of f(x): (-∞, ∞)
Next, let’s consider the function g(x) = 45x + 6. This function is essentially the same as f(x) but shifted upwards by 6 units. This vertical shift does not change the basic nature of the function’s output. As x still varies over all real numbers, g(x) will also cover all real numbers, but it will be shifted. When x approaches negative and positive infinity, g(x) approaches negative and positive infinity, respectively. Thus, the range of g(x) becomes:
- Range of g(x): (-∞, ∞)
In conclusion, both functions have the same range of all real numbers, represented as:
- Range of f(x): (-∞, ∞)
- Range of g(x): (-∞, ∞)