Final answer:
The transformation represented by g(x)=4^(0.5x-5) involves a horizontal compression by a factor of 2 and a vertical translation downward by 5 units.
Step-by-step explanation:
The transformation of f(x)=4x represented by g(x)=40.5x-5 includes a horizontal compression and a vertical translation. The coefficient 0.5 in the exponent of the function g(x) compresses the graph of f(x) horizontally by a factor of 2 (since 1/0.5 = 2), when looking at the reciprocal of the coefficient. Additionally, the -5 subtracted from all the exponents in g(x) translates the graph of f(x) down by 5 units.
To summarize these effects:
- The 0.5 in the exponent compresses the graph horizontally by a factor of 2.
- The -5 in the exponent translates the graph downward by 5 units.
This can be considered similar to how raising a number to a fractional power, like x2 = √x, denotes taking a square root, as seen in the equation provided. Here, however, the fraction is applied to the variable exponent, affecting how the graph of the exponential function stretches or shrinks horizontally.