The function you provided is \( y = \sqrt{3x - 7} + 2 \). To determine the coordinates of the point farthest to the left on the graph, we first need to establish the domain of the function.
The expression under the square root, \( 3x - 7 \), must be non-negative for the square root to be defined:
\[ 3x - 7 \geq 0 \]
Solving this inequality:
\[ 3x \geq 7 \] \[ x \geq \frac{7}{3} \]
The smallest x-coordinate allowed in the domain is \( x = \frac{7}{3} \).
Next, we find the corresponding y-coordinate by substituting \( x = \frac{7}{3} \) into the function:
\[ y = \sqrt{3\left(\frac{7}{3}\right) - 7} + 2 \]
Calculating inside the square root:
\[ 3 \cdot \frac{7}{3} - 7 = 7 - 7 = 0 \]
So, we have:
\[ y = \sqrt{0} + 2 = 0 + 2 = 2 \]
Thus, the coordinates of the point farthest to the left on the graph of the function are:
\[ \left( \frac{7}{3}, 2 \right) \]
In summary:
- The x-coordinate is \( \frac{7}{3} \).
- The y-coordinate is \( 2 \).
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