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Question: Restrict the domain of f(x)=3x^2+9x+12 so that the inverse is a function.
To ensure the inverse of a function is also a function, you need to make sure the original function is a one-to-one function. This means it has to pass the horizontal line test (any horizontal line will intersect the function at most once) in addition to the vertical line test. The function f(x) = 3x^2 + 9x + 12 is a quadratic function, and its graph is a parabola. By default, parabolas do not pass the horizontal line test since they will intersect a horizontal line twice, on either side of the vertex. However, we can restrict the domain so that the function only includes the values on one side of the vertex. For this function, the vertex is at x = -b/2a, from the formula of a quadratic function ax^2 +bx + c. Here a = 3, and b = 9, so the vertex is at x = -9/2(3) = -1.5. The parabola opens upwards since a>0. To make it a one-to-one function, you can restrict the domain to x>=-1.5 or x<=-1.5. The former will give you the right-hand side of the parabola (increasing function) and the latter will give you the left-hand side of the parabola (decreasing function). So, you can restrict the domain to either (-∞, -1.5] or [-1.5, ∞) depending on which part of the graph you want to keep for your function.
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