Answer :
To find a portion of the domain where the function [tex]\( f(x) = (x + 7)^2 - 6 \)[/tex] is one-to-one and determine its inverse function, follow these steps:
Step 1: Analyze the function [tex]\( f(x) = (x + 7)^2 - 6 \)[/tex]
The given function is a quadratic function, which means its graph is a parabola. Quadratic functions are not one-to-one over their entire domain because they are symmetric about their vertex, having one portion where they increase and another where they decrease.
Step 2: Determine the vertex of the parabola
The vertex form of a quadratic function [tex]\( a(x - h)^2 + k \)[/tex] indicates:
- [tex]\( h \)[/tex] is the x-coordinate of the vertex.
- [tex]\( k \)[/tex] is the y-coordinate of the vertex.
For the function [tex]\( f(x) = (x + 7)^2 - 6 \)[/tex]:
- The vertex is at [tex]\( x = -7 \)[/tex], since [tex]\( h = -7 \)[/tex].
- The y-coordinate at the vertex is [tex]\( -6 \)[/tex], since [tex]\( k = -6 \)[/tex].
So, the vertex of this function is [tex]\((-7, -6)\)[/tex].
Step 3: Restrict the domain to make the function one-to-one
Since the function is a parabola opening upwards (positive coefficient of [tex]\((x + 7)^2\)[/tex]), it is increasing for [tex]\(x \geq -7\)[/tex]. To make the function one-to-one, we restrict the domain to the portion where the function is increasing, starting from the vertex to [tex]\( \infty \)[/tex].
So, the restricted domain is [tex]\([-7, \infty)\)[/tex].
Step 4: Find the inverse function
To find the inverse function, follow these steps:
1. Start with the equation:
[tex]\[ y = (x + 7)^2 - 6 \][/tex]
2. Solve this equation for [tex]\( x \)[/tex]:
[tex]\[ y + 6 = (x + 7)^2 \][/tex]
[tex]\[ x + 7 = \pm \sqrt{y + 6} \][/tex]
3. Because we are considering the domain [tex]\( x \geq -7 \)[/tex], the [tex]\( \pm \)[/tex] square root takes the positive branch:
[tex]\[ x + 7 = \sqrt{y + 6} \][/tex]
4. Isolate [tex]\( x \)[/tex]:
[tex]\[ x = -7 + \sqrt{y + 6} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = -7 + \sqrt{x + 6} \][/tex]
Summary:
- The restricted domain for [tex]\( f \)[/tex] is [tex]\([-7, \infty)\)[/tex].
- The inverse function is [tex]\( f^{-1}(x) = -7 + \sqrt{x + 6} \)[/tex].
Therefore, the correct answer is:
The restricted domain for [tex]\( f \)[/tex] is [tex]\([-7, \infty)\)[/tex] and [tex]\( f^{-1}(x) = -7 + \sqrt{x + 6} \)[/tex].
Step 1: Analyze the function [tex]\( f(x) = (x + 7)^2 - 6 \)[/tex]
The given function is a quadratic function, which means its graph is a parabola. Quadratic functions are not one-to-one over their entire domain because they are symmetric about their vertex, having one portion where they increase and another where they decrease.
Step 2: Determine the vertex of the parabola
The vertex form of a quadratic function [tex]\( a(x - h)^2 + k \)[/tex] indicates:
- [tex]\( h \)[/tex] is the x-coordinate of the vertex.
- [tex]\( k \)[/tex] is the y-coordinate of the vertex.
For the function [tex]\( f(x) = (x + 7)^2 - 6 \)[/tex]:
- The vertex is at [tex]\( x = -7 \)[/tex], since [tex]\( h = -7 \)[/tex].
- The y-coordinate at the vertex is [tex]\( -6 \)[/tex], since [tex]\( k = -6 \)[/tex].
So, the vertex of this function is [tex]\((-7, -6)\)[/tex].
Step 3: Restrict the domain to make the function one-to-one
Since the function is a parabola opening upwards (positive coefficient of [tex]\((x + 7)^2\)[/tex]), it is increasing for [tex]\(x \geq -7\)[/tex]. To make the function one-to-one, we restrict the domain to the portion where the function is increasing, starting from the vertex to [tex]\( \infty \)[/tex].
So, the restricted domain is [tex]\([-7, \infty)\)[/tex].
Step 4: Find the inverse function
To find the inverse function, follow these steps:
1. Start with the equation:
[tex]\[ y = (x + 7)^2 - 6 \][/tex]
2. Solve this equation for [tex]\( x \)[/tex]:
[tex]\[ y + 6 = (x + 7)^2 \][/tex]
[tex]\[ x + 7 = \pm \sqrt{y + 6} \][/tex]
3. Because we are considering the domain [tex]\( x \geq -7 \)[/tex], the [tex]\( \pm \)[/tex] square root takes the positive branch:
[tex]\[ x + 7 = \sqrt{y + 6} \][/tex]
4. Isolate [tex]\( x \)[/tex]:
[tex]\[ x = -7 + \sqrt{y + 6} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = -7 + \sqrt{x + 6} \][/tex]
Summary:
- The restricted domain for [tex]\( f \)[/tex] is [tex]\([-7, \infty)\)[/tex].
- The inverse function is [tex]\( f^{-1}(x) = -7 + \sqrt{x + 6} \)[/tex].
Therefore, the correct answer is:
The restricted domain for [tex]\( f \)[/tex] is [tex]\([-7, \infty)\)[/tex] and [tex]\( f^{-1}(x) = -7 + \sqrt{x + 6} \)[/tex].
