Question

problems #13-16: sketch the graph of the inverse function for each function shown. state the domain for each.
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problems #17-20: use a graphing utility to graph each function and use the horizontal line test to determine whether the function is one-to-one and has an inverse.

1. $f(x) = \frac{1}{2}(x + 3)^2 - 4$
2. $g(x) = \frac{x - 2}{3x^2}$

$(x) = -x\sqrt{9 - x^2}$

1. $f(x) = \sqrt3{x - 4}$

Explanation:

Step1: Analyze the function type

The function \( f(x)=\sqrt[3]{x - 4} \) is a cube - root function. The general form of a cube - root function is \( y = \sqrt[3]{x - h}+k \), in this case, \( h = 4 \) and \( k = 0 \). The domain of the cube - root function \( y=\sqrt[3]{u} \) is all real numbers, because we can take the cube - root of any real number (positive, negative, or zero). For the function \( f(x)=\sqrt[3]{x - 4} \), the expression inside the cube - root is \( x - 4 \), and \( x-4 \) can be any real number as \( x \) varies over all real numbers.

Step2: Graph the function (conceptually)

The graph of \( y=\sqrt[3]{x} \) has a point - symmetric shape about the origin. The graph of \( y=\sqrt[3]{x - 4} \) is the graph of \( y = \sqrt[3]{x} \) shifted 4 units to the right.

Step3: Apply the Horizontal Line Test

To determine if a function is one - to - one (and thus has an inverse), we use the Horizontal Line Test. For a cube - root function \( y=\sqrt[3]{x - 4} \), assume we have a horizontal line \( y = k \), where \( k \) is a real number. If we solve the equation \( k=\sqrt[3]{x - 4} \) for \( x \), we get \( x=k^{3}+4 \). For each value of \( k \), there is exactly one value of \( x \). This means that no horizontal line will intersect the graph of \( f(x)=\sqrt[3]{x - 4} \) more than once. So the function \( f(x)=\sqrt[3]{x - 4} \) passes the Horizontal Line Test.

Step4: Conclusion about the inverse

Since the function \( f(x)=\sqrt[3]{x - 4} \) passes the Horizontal Line Test, it is one - to - one, and therefore it has an inverse function.

Answer:

The function \( f(x)=\sqrt[3]{x - 4} \) is one - to - one and has an inverse. The domain of \( f(x) \) is all real numbers (\( (-\infty,\infty) \)).