A function f is defined by f(x)=x^2+4
Evaluate f(3),f(-2),f(x+h) and find the domain and range of f
To evaluate f(3), we substitute x = 3 into the function f(x)=x^2+4:
f(3) = 3^2 + 4 = 9 + 4 = 13.
To evaluate f(-2), we substitute x = -2 into the function f(x)=x^2+4:
f(-2) = (-2)^2 + 4 = 4 + 4 = 8.
To evaluate f(x+h), we substitute x + h into the function f(x)=x^2+4:
f(x+h) = (x + h)^2 + 4.
The domain of a function is the set of all possible values of x in the function. Since the function f(x)=x^2+4 is a quadratic function, it is defined for all real values of x. Therefore, the domain of f is all real numbers.
The range of a function is the set of all possible values of f(x) for the given domain. For the function f(x)=x^2+4, the minimum value of x^2 is 0 (for x = 0), so the minimum value of f(x) is 4. Therefore, the range of f is all real numbers greater than or equal to 4.
To evaluate the function f(x), we substitute the given values of x into the function.
1. To find f(3):
f(3) = (3)^2 + 4
= 9 + 4
= 13
So, f(3) = 13.
2. To find f(-2):
f(-2) = (-2)^2 + 4
= 4 + 4
= 8
So, f(-2) = 8.
3. To find f(x+h):
f(x+h) = (x+h)^2 + 4
= x^2 + 2*x*h + h^2 + 4
Domain is the set of all possible inputs (x-values) for the function. Since f(x) is a polynomial, the domain is all real numbers (-∞, +∞).
Range is the set of all possible outputs (y-values) of the function. In this case, the function is a quadratic function (x^2 + 4), and since the leading coefficient is positive (1), the graph of the function will open upwards. Therefore, the minimum value of the function is at the vertex, which occurs at x = 0. The function increases as x moves away from 0, so the range is all real numbers greater than or equal to the minimum value, f(0) = 4. So, the range is [4, ∞).
To evaluate the function at different values, you simply substitute the given value for x in the function equation and perform the calculations. Let's evaluate f(3):
f(3) = (3)^2 + 4
f(3) = 9 + 4
f(3) = 13
So, f(3) = 13.
Similarly, let's evaluate f(-2):
f(-2) = (-2)^2 + 4
f(-2) = 4 + 4
f(-2) = 8
So, f(-2) = 8.
Now, let's evaluate f(x+h):
f(x+h) = (x+h)^2 + 4
f(x+h) = x^2 + 2xh + h^2 + 4
The domain of a function defines the set of all possible input values. Since the function f(x) = x^2 + 4 is a polynomial function, there are no particular restrictions on the domain. Therefore, the domain of f is all real numbers, (-∞, ∞).
The range of a function defines the set of all possible output values. In this case, the function f(x) = x^2 + 4 represents a quadratic equation where the coefficient of x^2 is positive, which means that the parabola opens upwards. Therefore, the minimum value the function can have is 4. Consequently, the range of f is all real numbers greater than or equal to 4, [4, ∞).