To determine the ranges of the functions f(x) = 45x and g(x) = 45x + 6, let’s analyze each function one by one.

1. Range of f(x) = 45x:

• This function is a linear function with a slope of 45. Linear functions are defined for all real numbers, which means the output can take any real value.
• As x ranges from negative infinity to positive infinity, f(x) will also range from negative infinity to positive infinity.

Thus, the range of f(x) = 45x is: R = (-∞, ∞).

2. Range of g(x) = 45x + 6:

• Just like with f(x), g(x) is also a linear function (with the same slope of 45), but it has a y-intercept of 6, shifting the entire graph upward by 6 units.
• While the slope remains the same, the addition of 6 means that as x goes from negative infinity to positive infinity, g(x) will also range infinitely but starting from 6.

Therefore, the range of g(x) = 45x + 6 is: R = (6 – ∞, ∞), or simply: R = (−∞, ∞).

Conclusion:

• Both functions f(x) = 45x and g(x) = 45x + 6 have the same range, which encompasses all real numbers.

Hence, the ranges of both functions can be succinctly expressed as: R = (-∞, ∞).