Given the function f(x) = (x - 1)^2 with a domain of x >= 7, find the range of f.
Understand the Problem
The question is asking to find the range of the function f(x) = (x - 1)^2, given that the domain of f is all values of x where x >= 7. This means we need to determine the set of all possible output values (f(x)) of the function when the input values (x) are greater than or equal to 7.
To solve this, we should first consider the vertex of the parabola defined by f(x) and then evaluate the function at the lower bound of the domain (x = 7) to find the minimum value of f(x) within the given domain. Since the parabola opens upwards, the range will be all values greater than or equal to this minimum value.
Answer
$f(x) \geq 36$
Answer for screen readers
$f(x) \geq 36$
Steps to Solve
- Find the vertex of the parabola
The function is given as $f(x) = (x - 1)^2$. This is a parabola in vertex form, $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In this case, $h = 1$ and $k = 0$, so the vertex is at $(1, 0)$.
- Determine if the parabola opens upwards or downwards
Since the coefficient $a$ in front of the $(x - 1)^2$ term is positive (specifically, $a = 1$), the parabola opens upwards.
- Consider the given domain restriction
The domain is restricted to $x \geq 7$. Since the vertex of the parabola is at $x = 1$, the function is increasing for all $x \geq 1$. Therefore, over the domain $x \geq 7$, the minimum value of the function will occur at $x = 7$.
- Evaluate the function at the lower bound of the domain
Substitute $x = 7$ into the function: $f(7) = (7 - 1)^2 = (6)^2 = 36$
- Determine the range
Since the parabola opens upwards and the domain is $x \geq 7$, the minimum value of the function is 36 at $x = 7$. Therefore, the range of the function is all values greater than or equal to 36.
$f(x) \geq 36$
More Information
The range represents all possible output values of the function for the given domain. In this case, because the domain is restricted to $x \geq 7$, the function only takes on values greater than or equal to 36.
Tips
A common mistake is to assume that the range is all values greater than or equal to zero, because $(x-1)^2$ is always non-negative. However, the domain restriction $x \geq 7$ means that the function never actually achieves its minimum value of zero within the given domain.
AI-generated content may contain errors. Please verify critical information
