How to find the domain of the inverse function of x^2-10x+25
f(x) = x^2-10x+25
The domain of the inverse of f is the same as the range of f.
To find the domain of the inverse function of \(f(x) = x^2-10x+25\), we need to determine the range of the original function.
Step 1: Find the inverse function
To find the inverse function, we interchange the dependent and independent variables. Let's denote the inverse function as \(f^{-1}(x)\) and solve for \(x\).
\(y = x^2-10x+25\)
Swap \(y\) and \(x\):
\(x = y^2-10y+25\)
Rearrange to solve for \(y\):
\(y^2-10y+25 = x\)
Now, solve for \(y\) using the quadratic formula:
\(y = \frac{-(-10) \pm \sqrt{(-10)^2-4(1)(25)}}{2(1)}\)
\(y = \frac{10 \pm \sqrt{100-100}}{2}\)
\(y = \frac{10 \pm \sqrt{0}}{2}\)
\(y = \frac{10 \pm 0}{2}\)
\(y = 5\)
Therefore, we have found that the inverse function is \(f^{-1}(x) = 5\).
Step 2: Find the domain of the inverse function
The domain of the inverse function is the range of the original function, \(f(x) = x^2-10x+25\), which is the set of all possible values that \(f(x)\) can take.
From analyzing the original function, we can see that it is a quadratic equation, and the graph of a quadratic equation is a parabola. In this case, the parabola opens upwards since the coefficient of \(x^2\) is positive. Thus, the range of \(f(x)\) is all values greater than or equal to the vertex.
To find the vertex, we can use the formula \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\), and \(b\) is the coefficient of \(x\). In this case, \(a = 1\) and \(b = -10\).
\(x = -\frac{-10}{2(1)}\)
\(x = \frac{10}{2}\)
\(x = 5\)
So, the vertex of the parabola is at \(x = 5\). Since the parabola opens upwards, the range of \(f(x)\) is all values greater than or equal to the y-coordinate of the vertex. In this case, the y-coordinate is also 5. Therefore, the range of \(f(x)\) is \(y \geq 5\). Since this is the range of \(f(x)\), it will be the domain of \(f^{-1}(x)\).
In conclusion, the domain of the inverse function of \(f(x) = x^2-10x+25\) is \(x \geq 5\).
To find the domain of the inverse function of a given function, you need to determine the range of the original function. This is because the domain of the inverse function corresponds to the range of the original function.
In this case, the given function is f(x) = x^2 - 10x + 25.
To find the range of this function, you can consider its graph or use calculus. By analyzing the graph or finding the minimum/maximum points using calculus techniques, you can determine that the range of f(x) is [0, infinity). This means that all values greater than or equal to zero are included in the range.
To find the domain of the inverse function, you need to consider this range as the set of possible inputs for the inverse function. Therefore, the domain of the inverse function is [0, infinity).