state the domain and range of the following
f(x)= x squared +4, x more than or equal to zero
f(x)= 4/x+2
To find the domain and range of the given functions, we need to identify any restrictions on the input (x) and determine the possible output (y) values.
For the first function, f(x) = x^2 + 4, x is specified to be more than or equal to zero. It means that the function is only defined for x values that are greater than or equal to zero.
Domain (input): x ≥ 0
Next, let's analyze the second function, f(x) = 4/(x + 2). In this case, we need to look for any restrictions on x values that would make the denominator equal to zero since division by zero is undefined. Considering this, we set the denominator (x + 2) to be zero and solve for x.
x + 2 = 0
x = -2
Hence, x cannot be equal to -2. Therefore, the domain of the second function would be all real numbers except -2.
Domain (input): x ≠ -2
For both functions, the range (output) is the set of all possible y-values that the function can produce. Since the first function is a quadratic function in the form f(x) = x^2 + 4, it opens upward since the coefficient of x^2 is positive (+1). Therefore, the minimum value of y occurs at the vertex, and it is 4.
Range (output): y ≥ 4
For the second function f(x) = 4/(x + 2), as the x-values approach -2 from both sides, f(x) will tend towards positive or negative infinity, depending on the direction of approach. Hence, the range for the second function is all real numbers except zero.
Range (output): y ≠ 0
In summary:
Function 1:
Domain: x ≥ 0
Range: y ≥ 4
Function 2:
Domain: x ≠ -2
Range: y ≠ 0