Mathematics

The key is recognizing how a horizontal scaling and vertical shift transform both the domain and the average value. When you compress the input by a factor of 4 (via $4x$), you're "speeding up" through the original function, but the average value of $f$ itself doesn't change—you're just sampling the same function values. The constant $D$ shifts all outputs uniformly, so it adds directly to the average. Set up the average value formula for $g$ on $[0, \tfrac{1}{2}]$ and use the substitution $u = 4x$ to connect it back to $f$'s known average on $[0,2]$.

The key is recognizing how a horizontal scaling and vertical shift transform both the domain and the average value. When you compress the input by a factor of 4 (via $4x$), you're "speeding up" through the original function, but the average value of $f$ itself doesn't change—you're just sampling the same function values. The constant $D$ shifts all outputs uniformly, so it adds directly to the average. Set up the average value formula for $g$ on $[0, \tfrac{1}{2}]$ and use the substitution $u = 4x$ to connect it back to $f$'s known average on $[0,2]$.