Question

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Explanation:

Step1: Recall Inverse Function Graph Property

The graph of an inverse function of \( y = f(x) \) is the reflection of the graph of \( y = f(x) \) over the line \( y = x \). Also, we can analyze the type of function (linear, square root, cubic, rational) to match with their inverse graphs.

Step2: Analyze Each Function

• Function \( f(x)=\sqrt{x - 1}-3 \): The original function is a square - root function (domain \( x\geq1 \), range \( y\geq - 3 \)). Its inverse will be a quadratic - like function (after solving \( x=\sqrt{y - 1}-3\Rightarrow x + 3=\sqrt{y - 1}\Rightarrow(y - 1)=(x + 3)^2\Rightarrow y=(x + 3)^2+1 \), which is a parabola opening upwards with vertex at \( (-3,1) \). The first graph (top - most) has a curve that seems to be a parabola - like (reflection of square - root), so it matches \( f(x)=\sqrt{x - 1}-3 \).
• Function \( f(x)=x^{3}+5 \): The original function is a cubic function (one - to - one, since the derivative \( f^\prime(x)=3x^{2}\geq0 \) and only zero at \( x = 0 \), so strictly increasing). Its inverse will also be a cubic - like function. The fourth graph (bottom - most) has a cubic - like curve, so it matches \( f(x)=x^{3}+5 \).
• Function \( f(x)=\frac{1}{2}x \): This is a linear function with slope \( \frac{1}{2} \). Its inverse will be \( y = 2x \) (solving \( x=\frac{1}{2}y\Rightarrow y = 2x \)), which is a linear function with slope 2. The second graph (second from top) is a linear graph, so it matches \( f(x)=\frac{1}{2}x \).
• Function \( f(x)=\frac{6}{x} \): This is a rational function (hyperbola). Its inverse is itself (since if \( x=\frac{6}{y}\Rightarrow y=\frac{6}{x} \)). The third graph (third from top) is a hyperbola, so it matches \( f(x)=\frac{6}{x} \).

Answer:

• Top graph (first graph of inverse function) : \( f(x)=\sqrt{x - 1}-3 \)
• Second graph (second from top) : \( f(x)=\frac{1}{2}x \)
• Third graph (third from top) : \( f(x)=\frac{6}{x} \)
• Fourth graph (bottom graph) : \( f(x)=x^{3}+5 \)