Final answer:
The domain of the function is all real numbers, and the range is (-∞, 9].
Step-by-step explanation:
The domain of a function represents the set of all possible input values (x-values) that the function can accept. To find the domain of the given function, we need to determine any restrictions on the input values.
f(x) = -4x² + 8x + 5
Since this is a quadratic function, it is defined for all real numbers. Therefore, the domain is all real numbers (-∞, ∞).
The range of a function represents the set of all possible output values (y-values) that the function can produce. To find the range of the given function, we need to consider the shape of the quadratic function.
The coefficient of x² is negative (-4), which means the graph of the function opens downwards and has a maximum point. Since the leading coefficient is negative, the range is limited to the y-values that are less than or equal to the y-coordinate of the maximum point. To find the y-coordinate of the maximum point, we can use the formula x = -b/2a.
In this case, a = -4 and b = 8, so the x-coordinate of the maximum point is x = -8/(2*(-4)) = 1. Substituting this x-value back into the function, we get f(1) = -4(1)² + 8(1) + 5 = 9. Therefore, the range of the function is (-∞, 9].