Please draw graph show all calculations and co ordinate

Okay, sthamntungwa, I can't draw the graph for you directly, but I will provide all the necessary calculations and coordinates, along with detailed instructions on how to draw it accurately.

First, let's summarize the functions and their domains/ranges:

• $f(x) = 2x - 5$
• $f^{-1}(x) = \frac{1}{2}x + \frac{5}{2}$
• Domain of $f^{-1}(x)$: $\{x: -5 < x \le 7, x \in \mathbb{R}\}$
• Range of $f(x)$: $\{y: -5 < y \le 7, y \in \mathbb{R}\}$
• Domain of $f(x)$: $\{x: 0 < x \le 6, x \in \mathbb{R}\}$ (This is the range of $f^{-1}(x)$)
• Range of $f^{-1}(x)$: $\{y: 0 < y \le 6, y \in \mathbb{R}\}$ (This is the domain of $f(x)$)

Calculations and Coordinates for Graphing

1. Line of Reflection: $y=x$

This line passes through the origin.

• A key point on this line is $\boxed{\mathbf{(0,0)}}$.
• Another point could be $\boxed{\mathbf{(1,1)}}$.

2. Graph of $f(x) = 2x - 5$

The domain for $f(x)$ is $\{x: 0 < x \le 6, x \in \mathbb{R}\}$. We need to calculate the $y$-values at the endpoints of this domain.

• Calculation for $x=0$ (open endpoint):

  $$f(0) = 2(0) - 5 = -5$$

  The coordinate is $\boxed{\mathbf{(0, -5)}}$. This point should be an open circle on your graph because $x > 0$.

• Calculation for $x=6$ (closed endpoint):

  $$f(6) = 2(6) - 5 = 12 - 5 = 7$$

  The coordinate is $\boxed{\mathbf{(6, 7)}}$. This point should be a closed circle on your graph because $x \le 6$.

3. Graph of $f^{-1}(x) = \frac{1}{2}x + \frac{5}{2}$

The domain for $f^{-1}(x)$ is given as $\{x: -5 < x \le 7, x \in \mathbb{R}\}$. We need to calculate the $y$-values at the endpoints of this domain.

• Calculation for $x=-5$ (open endpoint):

  $$f^{-1}(-5) = \frac{1}{2}(-5) + \frac{5}{2} = -\frac{5}{2} + \frac{5}{2} = 0$$

  The coordinate is $\boxed{\mathbf{(-5, 0)}}$. This point should be an open circle on your graph because $x > -5$.

• Calculation for $x=7$ (closed endpoint):

  $$f^{-1}(7) = \frac{1}{2}(7) + \frac{5}{2} = \frac{7}{2} + \frac{5}{2} = \frac{12}{2} = 6$$

  The coordinate is $\boxed{\mathbf{(7, 6)}}$. This point should be a closed circle on your graph because $x \le 7$.

Instructions for Drawing the Graph

1. Draw Axes: Draw a clear Cartesian coordinate system with labeled x-axis and y-axis. Make sure your scale is appropriate to fit all points (from -5 to 7 on x-axis, and -5 to 7 on y-axis).
2. Draw $y=x$: Draw a straight line passing through $(0,0)$ and $(1,1)$ (or any other point where $x=y$). Label this line as $y=x$.
3. Plot $f(x)$:
   • Place an open circle at $\mathbf{(0, -5)}$.
   • Place a closed circle at $\mathbf{(6, 7)}$.
   • Draw a straight line segment connecting these two points.
   • Label this line segment as $f(x)$.
4. Plot $f^{-1}(x)$:
   • Place an open circle at $\mathbf{(-5, 0)}$.
   • Place a closed circle at $\mathbf{(7, 6)}$.
   • Draw a straight line segment connecting these two points.
   • Label this line segment as $f^{-1}(x)$.

Ensure all labels ($f(x)$, $f^{-1}(x)$, $y=x$, x-axis, y-axis) are clearly visible.