Questions: Given the function f(x)=(x+8)^3, complete parts a through c. (a) Find an equation for f^(-1)(x). (b) Graph f and f^(-1) in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of f and f^(-1). (c) State the domain and range of f and f^(-1) using interval notation. The domain of f(x) is , and the range of f(x) is .

Given the function f(x)=(x+8)^3, complete parts a through c.
(a) Find an equation for f^(-1)(x).
(b) Graph f and f^(-1) in the same rectangular coordinate system.
(c) Use interval notation to give the domain and the range of f and f^(-1).
(c) State the domain and range of f and f^(-1) using interval notation.

The domain of f(x) is , and the range of f(x) is .
Transcript text: Given the function $f(x)=(x+8)^{3}$, complete parts a through $c$. (a) Find an equation for $f^{-1}(x)$. (b) Graph $f$ and $f^{-1}$ in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of $f$ and $f^{-1}$. (c) State the domain and range of $f$ and $f^{-1}$ using interval notation. The domain of $f(x)$ is $\square$ , and the range of $f(x)$ is $\square$ .
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Solution

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Solution Steps

Step 1: Find an equation for f1(x) f^{-1}(x)

Given the function f(x)=(x+8)3 f(x) = (x + 8)^3 , we need to find its inverse f1(x) f^{-1}(x) .

  1. Start by setting y=f(x) y = f(x) : y=(x+8)3 y = (x + 8)^3

  2. Solve for x x in terms of y y : y=(x+8)3    y3=x+8    x=y38 y = (x + 8)^3 \implies \sqrt[3]{y} = x + 8 \implies x = \sqrt[3]{y} - 8

  3. Swap x x and y y to get the inverse function: f1(x)=x38 f^{-1}(x) = \sqrt[3]{x} - 8

Step 2: Graph f f and f1 f^{-1} in the same rectangular coordinate system

To graph f(x)=(x+8)3 f(x) = (x + 8)^3 and f1(x)=x38 f^{-1}(x) = \sqrt[3]{x} - 8 :

  1. Plot the function f(x)=(x+8)3 f(x) = (x + 8)^3 :

    • This is a cubic function shifted 8 units to the left.
  2. Plot the inverse function f1(x)=x38 f^{-1}(x) = \sqrt[3]{x} - 8 :

    • This is a cube root function shifted 8 units down.
  3. Ensure the graphs are reflections of each other across the line y=x y = x .

Step 3: Use interval notation to give the domain and range of f f and f1 f^{-1}
  1. Determine the domain and range of f(x)=(x+8)3 f(x) = (x + 8)^3 :

    • The domain of f(x) f(x) is all real numbers (,) (-\infty, \infty) .
    • The range of f(x) f(x) is all real numbers (,) (-\infty, \infty) .
  2. Determine the domain and range of f1(x)=x38 f^{-1}(x) = \sqrt[3]{x} - 8 :

    • The domain of f1(x) f^{-1}(x) is all real numbers (,) (-\infty, \infty) .
    • The range of f1(x) f^{-1}(x) is all real numbers (,) (-\infty, \infty) .

Final Answer

  1. The inverse function is f1(x)=x38 f^{-1}(x) = \sqrt[3]{x} - 8 .
  2. The domain and range of f(x) f(x) are (,) (-\infty, \infty) .
  3. The domain and range of f1(x) f^{-1}(x) are (,) (-\infty, \infty) .
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