The function \(f(x) = 2x + 5\) is a linear function with a positive gradient of 2 and a \(y\)-intercept of 5.

To find the inverse function \(f^{-1}(x)\), we swap \(x\) and \(y\) in the equation \(y = 2x + 5\) and solve for \(y\):

\(x = 2y + 5\)

\(x - 5 = 2y\)

\(y = \frac{x - 5}{2}\)

Thus, \(f^{-1}(x) = \frac{x - 5}{2}\), which is also a linear function with a positive gradient of \(\frac{1}{2}\) and a \(y\)-intercept of \(-\frac{5}{2}\).

The line \(y = x\) is the line of symmetry between \(y = f(x)\) and \(y = f^{-1}(x)\), indicating that the graphs are reflections of each other across this line.

The function \(f(x) = 2x + 3\) is defined for \(x \leq 0\). To find the inverse, solve \(y = 2x + 3\) for \(x\):

\(y - 3 = 2x\)

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

Thus, \(f^{-1}(x) = \frac{x - 3}{2}\).

The graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are reflections of each other in the line \(y = x\).

To find the intersection, set \(f(x) = f^{-1}(x)\):

\(2x + 3 = \frac{x - 3}{2}\)

Multiply through by 2 to clear the fraction:

\(4x + 6 = x - 3\)

\(3x = -9\)

\(x = -3\)

Substitute \(x = -3\) back into \(f(x)\):

\(y = 2(-3) + 3 = -3\)

Thus, the point of intersection is \((-3, -3)\).

The function given is \(f(x) = 3x - 4\).

To find the inverse, solve for \(x\) in terms of \(y\):

\(y = 3x - 4\)

\(y + 4 = 3x\)

\(x = \frac{y + 4}{3}\)

Thus, \(f^{-1}(x) = \frac{x + 4}{3}\).

The graph of \(y = f(x) = 3x - 4\) is a straight line with a slope of 3 and a y-intercept of -4.

The graph of \(y = f^{-1}(x) = \frac{x + 4}{3}\) is a straight line with a slope of \(\frac{1}{3}\) and a y-intercept of \(\frac{4}{3}\).

The line \(y = x\) is the line of symmetry between the function and its inverse.

According to the mark scheme, \(y = f(x)\) is correct in the 1st and 4th quadrants, \(y = f^{-1}(x)\) is correct in the 1st and 2nd quadrants, and \(y = x\) is marked or quoted.

The function given is \(f(x) = 2x + 1\).

To find the inverse function \(f^{-1}(x)\), we set \(y = 2x + 1\) and solve for \(x\):

\(y = 2x + 1\)

\(y - 1 = 2x\)

\(x = \frac{1}{2}(y - 1)\)

Thus, \(f^{-1}(x) = \frac{1}{2}(x - 1)\).

The graph of \(y = f(x) = 2x + 1\) is a straight line with a slope of 2, starting from \((0, 1)\).

The graph of \(y = f^{-1}(x) = \frac{1}{2}(x - 1)\) is a straight line with a slope of \(\frac{1}{2}\), starting from \((1, 0)\).

These two graphs are reflections of each other across the line \(y = x\).

The function \(f(x) = 3x - 2\) is a linear function with a slope of 3 and a y-intercept of -2.

To find the inverse function \(f^{-1}(x)\), we swap \(x\) and \(y\) and solve for \(y\):

\(x = 3y - 2\)

\(x + 2 = 3y\)

\(y = \frac{1}{3}(x + 2)\)

Thus, \(f^{-1}(x) = \frac{1}{3}(x + 2)\).

The graph of \(y = f(x)\) is a line with slope 3 and y-intercept -2.

The graph of \(y = f^{-1}(x)\) is a line with slope \(\frac{1}{3}\) and y-intercept \(\frac{2}{3}\).

The line \(y = x\) is the line of symmetry, and the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are reflections of each other across this line.