write down the domain and the range of the following:
1)g(x)=-(x-1)^2 + 3
2)f(x)=1/x+2 - 5
To find the domain and range of a function, we need to understand the limitations on the input values (domain) and the output values (range).
1) Let's find the domain and range for the function g(x) = -(x-1)^2 + 3:
Domain:
The domain refers to all possible input values (x) for which the function g(x) is defined. In this case, since g(x) is a quadratic function, it will be defined for all real numbers.
Therefore, the domain of g(x) is (-∞, ∞) or (-infinity, infinity).
Range:
The range captures the set of all possible output values (y) that a function can produce. For a quadratic function of the form g(x) = a(x-h)^2 + k, where a, h, and k are constants, the vertex represents the highest or lowest point on the graph. Since the coefficient of the squared term in g(x) is negative (-1), the graph opens downward, and the vertex will represent the maximum value.
In this function, g(x) = -(x-1)^2 + 3, the vertex is (1, 3), and the graph is a downward-facing parabola. Therefore, the maximum value of y will be 3 (the y-coordinate of the vertex).
So, the range of g(x) will be (-∞, 3].
2) Let's find the domain and range for the function f(x) = 1/(x+2) - 5:
Domain:
The domain refers to all possible input values (x) for which the function f(x) is defined. In this case, the function is a rational function. In a rational function, the denominator cannot be zero since dividing by zero is undefined. Therefore, the function f(x) will be defined for all real numbers except for x = -2.
Hence, the domain of f(x) is (-∞, -2) U (-2, ∞) or (-infinity, -2) union (-2, infinity).
Range:
The range represents all possible output values (y) produced by a function. In this case, as x approaches infinity or negative infinity, the absolute value of (x+2) gets larger, causing the absolute value of f(x) to approach zero. Since f(x) = 1/(x+2) - 5, the output value (y) can approach negative infinity but will always be at least 5 units less.
Therefore, the range of f(x) will be (-∞, -5).