The function g(x) is defined as g(x) = x² - 4x + 5 for all x. g(x) can be written as g(x) = (x - 2)² + 1. Write down the range of g(x).
Understand the Problem
The question defines a quadratic function g(x) and asks us to determine its range. We can find the range by recognizing that the given form g(x) = (x - 2)² + 1 reveals the vertex of the parabola, which allows us to quickly determine the minimum value of the function and thus its range.
Answer
$g(x) \ge 1$
Answer for screen readers
$g(x) \ge 1$
Steps to Solve
-
Identify the vertex form of the quadratic The given function is in vertex form: $g(x) = (x - 2)^2 + 1$.
-
Determine the vertex The vertex of the parabola is $(2, 1)$.
-
Determine the direction of opening Since the coefficient of the $(x - 2)^2$ term is positive (1), the parabola opens upwards. This means that the vertex represents the minimum point of the function.
-
Determine the range Since the parabola opens upwards and the vertex is at $(2, 1)$, the minimum value of $g(x)$ is 1. The function can take any value greater than or equal to 1. Therefore, the range of $g(x)$ is $g(x) \ge 1$.
$g(x) \ge 1$
More Information
The range represents all possible output values of a function. In this case, since the quadratic has a minimum value, the range includes all values greater than or equal to the minimum.
Tips
A common mistake is to look at the original form of the equation $g(x) = x^2 - 4x + 5$ and not recognize the significance of the vertex form $g(x) = (x - 2)^2 + 1$. The vertex form immediately gives the vertex $(2, 1)$ which helps to determine the minimum value and thus the range.
AI-generated content may contain errors. Please verify critical information
