Question

Find an explicit formula for \({f^{ - 1}}\) and use it to graph \({f^{ - 1}},f,\) and the line \(y = x\) on the same screen. To check your work, whether the graphs of \(f\) and \({f^{ - 1}}\) are reflections about the line.

31. \(f\left( x \right) = \sqrt {{\bf{4}}x + {\bf{3}}} \)

Short answer

According to the graph, we can see that \(f\) and \({f^{ - 1}}\) are reflections about the line \(y = x\).

Step-by-step solution

Condition to find the inverse function and graph of \({f^{ - 1}}\)

Step 1: Write the equation as \(y = f\left( x \right)\).

Step 2: If possible, solve the equation in terms of \(y\).

Step 3: Express the inverse \({f^{ - 1}}\) as afunction of \(x\), for thatinterchange\(x\)and \(y\). The equation becomes \(y = {f^{ - 1}}\left( x \right)\).

The graph of \({f^{ - 1}}\) is determined byreflecting the graph of \(f\) about theline \(y = x\).

Determine the explicit formula for \({f^{ - 1}}\) and use it to graph \({f^{ - 1}},f,\) and the line \(y = x\)

Write the equation as \(y = \sqrt {4x + 3} \left( {y \ge 0} \right)\) and solve the equation for \(x\) as shown below:

\(\begin{aligned}y &= \sqrt {4x + 3} \\{y^2} &= 4x + 3\\x &= \frac{{{y^2} - 3}}{4}\end{aligned}\)

Interchange \(x\) and \(y\) in the above equation as shown below:

\(y = \frac{{{x^2} - 3}}{4}\)

Therefore, the inverse function is \({f^{ - 1}}\left( x \right) = \frac{{{x^2} - 3}}{4}\left( {x \ge 0} \right)\).

Sketch the graph of \(y = \sqrt {4x + 3} \) and then we reflect about the line \(y = x\) to get the graph of \({f^{ - 1}}\).

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({\left( {4X + 3} \right)^{1/2}}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function \(f\left( x \right) = \sqrt {4x + 3} \) is shown below:

Thus, according to the graph, the functions \(f\) and \({f^{ - 1}}\) are reflections about the line \(y = x\).