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Response 1
PROMPT
If f(x)=−2x2−5xf(x)=−2x2−5x and g(x)=3x+2g(x)=3x+2, find the domains and ranges of f(x)f(x), g(x)g(x), and f(g(x))f(g(x))
Response 1
To determine the domains and ranges of the functions f(x)=−2x2−5xf(x)=−2x2−5x, g(x)=3x+2g(x)=3x+2, and the composite function f(g(x))f(g(x)), we'll address each function separately.
1. Domain and range of f(x)=−2x2−5xf(x)=−2x2−5x:
* Domain of f(x)f(x): f(x)f(x) is a quadratic function, which is defined for all real numbers. Hence, the domain is:
\text{Domain of } f(x): \mathbb{R}
* Range of f(x)f(x): Since f(x)=−2x2−5xf(x)=−2x2−5x is a downward-opening parabola (the coefficient of x2x2 is negative), it has a maximum value at the vertex. The x-coordinate of the vertex can be found using x=−b2ax=−2ab, where a=−2a=−2 and b=−5b=−5. Thus, the x-coordinate is:
x = -\frac{-5}{2(-2)} = \frac{5}{4}
Plug this back into the function to find the y-coordinate (maximum point):
f\left(\frac{5}{4}\right) = -2\left(\frac{5}{4}\right)^2 - 5\left(\frac{5}{4}\right)
= -2 \left(\frac{25}{16}\right) - \frac{25}{4}
= -\frac{50}{16} - \frac{100}{16} = -\frac{150}{16} = -\frac{75}{8}
Therefore, the range of f(x)f(x) is:
\text{Range of } f(x): \left(-\infty, -\frac{75}{8}\right]
2. Domain and range of g(x)=3x+2g(x)=3x+2:
* Domain of g(x)g(x): g(x)g(x) is a linear function, which is defined for all real numbers. Hence, the domain is:
\text{Domain of } g(x): \mathbb{R}
* Range of g(x)g(x): Since g(x)g(x) is a linear function, its range is also all real numbers. Hence, the range is:
\text{Range of } g(x): \mathbb{R}
3. Domain and range of the composite function f(g(x))f(g(x)):
The composite function is:
f(g(x)) = f(3x + 2) = -2(3x + 2)^2 - 5(3x + 2)
* Domain of f(g(x))f(g(x)): Since both f(x)f(x) and g(x)g(x) are defined for all real numbers, f(g(x))f(g(x)) is also defined for all real numbers:
\text{Domain of } f(g(x)): \mathbb{R}
* Range of f(g(x))f(g(x)): The form of f(g(x))f(g(x)) is still a quadratic expression in terms of xx after expanding and simplifying it. It will still open downwards like f(x)f(x), indicating that its range is structurally similar:
   * The quadratic will have its maximum value which corresponds to the maximum value computed from:
f(g(x)) = -2(3x + 2)^2 - 5(3x + 2) 
The parabola will have a maximum value similar in form to part 1, (−2x2−5x(−2x2−5x), thus its maximum will occur due to its vertex similarly.
Ultimately, the general behavior for the composite function follows as:
\text{Range of } f(g(x)): \left(-\infty, \text{some maximum value}$$
\]

So, the domains and ranges of the functions are:
- Domain of $$ f(x) $$: $$\mathbb{R}$$
- Range of $$ f(x) $$: $$(- \infty, -\frac{75}{8}]$$
- Domain of $$ g(x) $$: $$\mathbb{R}$$
- Range of $$ g(x) $$: $$\mathbb{R}$$
- Domain of $$ f(g(x)) $$: $$\mathbb{R}$$
- Range of $$ f(g(x)) $$: $$\left(-\infty, \text{maximum of the composite function} \right]$$, which can similarly be calculated as shown above for explicit values, being downwards opening quadratic.