To find the domain and range of the inverse function $f^{-1}(x) = x^2 - 2x + 6$, let's carefully analyze it step by step.

1. Domain of $f^{-1}(x)$:

The domain of a function refers to the set of all possible input values (i.e., $x$) for which the function is defined. In this case, $f^{-1}(x) = x^2 - 2x + 6$ is a quadratic function, which is defined for all real numbers.

• Domain: $(-\infty, \infty)$

2. Range of $f^{-1}(x)$:

The range refers to the set of all possible output values of the function. Since $f^{-1}(x)$ is a quadratic function with a positive leading coefficient (the coefficient of $x^2$ is 1), the graph of the function is a parabola opening upwards.

We can find the minimum value of the quadratic function by completing the square or using the vertex formula.

The vertex of a parabola $ax^2 + bx + c$ occurs at: $x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1$ Substitute $x = 1$ into the function to find the corresponding $y$-value: $f^{-1}(1) = (1)^2 - 2(1) + 6 = 1 - 2 + 6 = 5$ Thus, the minimum value of $f^{-1}(x)$ is 5, and since the parabola opens upwards, the range is from 5 to infinity.

• Range: $[5, \infty)$

Summary:

• Domain of $f^{-1}(x)$: $(-\infty, \infty)$
• Range of $f^{-1}(x)$: $[5, \infty)$

Would you like any further details on this explanation?

Related Questions:

1. What are the steps to graph a quadratic function like $f^{-1}(x) = x^2 - 2x + 6$?
2. How do you find the inverse of a function step by step?
3. How does completing the square help in finding the vertex of a quadratic function?
4. Can the range of a quadratic function ever be negative?
5. What are the domain and range of the original function $f(x)$, if we are given $f^{-1}(x)$?

Tip:

For any quadratic function of the form $ax^2 + bx + c$, the minimum or maximum value occurs at the vertex, given by $x = -\frac{b}{2a}$.