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Sagot :
To determine whether the function [tex]\( f(x) = x^3 - 1 \)[/tex] is one-to-one, we need to analyze its behavior and check if each output value [tex]\( y \)[/tex] has exactly one corresponding input value [tex]\( x \)[/tex].
### Step 1: Determine if the function is one-to-one
A function is one-to-one if and only if it passes both the horizontal line test and the vertical line test.
#### Vertical Line Test:
Any vertical line drawn through the graph of [tex]\( f(x) \)[/tex] should touch the graph at exactly one point. This is true for [tex]\( f(x) = x^3 - 1 \)[/tex] because it is a polynomial function of degree 3, which does not repeat any values for its input.
#### Horizontal Line Test:
Any horizontal line drawn through the graph of [tex]\( f(x) \)[/tex] should touch the graph at exactly one point. For [tex]\( f(x) = x^3 - 1 \)[/tex], since it is a monotonically increasing function (its derivative [tex]\( f'(x) = 3x^2 \)[/tex] is always non-negative and only equals zero at [tex]\( x = 0 \)[/tex]), it means that each value of [tex]\( f(x) \)[/tex] is achieved exactly once. Therefore, [tex]\( f(x) \)[/tex] passes the horizontal line test as well.
Since [tex]\( f(x) \)[/tex] passes both the vertical and horizontal line tests, the function is one-to-one.
### Step 2: Find the inverse function
To find the inverse function of [tex]\( f(x) = x^3 - 1 \)[/tex], follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 1 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y + 1 = x^3 \][/tex]
[tex]\[ x = (y + 1)^{1/3} \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = (x + 1)^{1/3} \][/tex]
Therefore, the equation for the inverse function is:
[tex]\[ y = f^{-1}(x) = (x + 1)^{1/3} \][/tex]
### Step 3: Graph [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex]
The graph of [tex]\( f(x) = x^3 - 1 \)[/tex] is a cubic curve shifted one unit down. The graph of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is the cube root function shifted one unit left. When graphed on the same axes, these functions will be symmetric with respect to the line [tex]\( y = x \)[/tex].
### Step 4: Determine the domain and range
- The domain of [tex]\( f(x) = x^3 - 1 \)[/tex] is all real numbers because you can cube any real number.
- The range of [tex]\( f(x) = x^3 - 1 \)[/tex] is all real numbers because the cubic function can produce any real number output.
- The domain of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is all real numbers because you can take the cube root of any real number.
- The range of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is all real numbers because the cube root function can produce any real number output.
### Summary
(a) The function [tex]\( f(x) \)[/tex] is one-to-one, and the inverse function is given by:
[tex]\[ f^{-1}(x) = (x + 1)^{1/3} \][/tex]
(b) The graph of [tex]\( f(x) = x^3 - 1 \)[/tex] and [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] should be symmetric with respect to the line [tex]\( y = x \)[/tex].
(c) The domain and range of [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex] are:
- Domain of [tex]\( f(x) \)[/tex]: All real numbers
- Range of [tex]\( f(x) \)[/tex]: All real numbers
- Domain of [tex]\( f^{-1}(x) \)[/tex]: All real numbers
- Range of [tex]\( f^{-1}(x) \)[/tex]: All real numbers
### Step 1: Determine if the function is one-to-one
A function is one-to-one if and only if it passes both the horizontal line test and the vertical line test.
#### Vertical Line Test:
Any vertical line drawn through the graph of [tex]\( f(x) \)[/tex] should touch the graph at exactly one point. This is true for [tex]\( f(x) = x^3 - 1 \)[/tex] because it is a polynomial function of degree 3, which does not repeat any values for its input.
#### Horizontal Line Test:
Any horizontal line drawn through the graph of [tex]\( f(x) \)[/tex] should touch the graph at exactly one point. For [tex]\( f(x) = x^3 - 1 \)[/tex], since it is a monotonically increasing function (its derivative [tex]\( f'(x) = 3x^2 \)[/tex] is always non-negative and only equals zero at [tex]\( x = 0 \)[/tex]), it means that each value of [tex]\( f(x) \)[/tex] is achieved exactly once. Therefore, [tex]\( f(x) \)[/tex] passes the horizontal line test as well.
Since [tex]\( f(x) \)[/tex] passes both the vertical and horizontal line tests, the function is one-to-one.
### Step 2: Find the inverse function
To find the inverse function of [tex]\( f(x) = x^3 - 1 \)[/tex], follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 1 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y + 1 = x^3 \][/tex]
[tex]\[ x = (y + 1)^{1/3} \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = (x + 1)^{1/3} \][/tex]
Therefore, the equation for the inverse function is:
[tex]\[ y = f^{-1}(x) = (x + 1)^{1/3} \][/tex]
### Step 3: Graph [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex]
The graph of [tex]\( f(x) = x^3 - 1 \)[/tex] is a cubic curve shifted one unit down. The graph of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is the cube root function shifted one unit left. When graphed on the same axes, these functions will be symmetric with respect to the line [tex]\( y = x \)[/tex].
### Step 4: Determine the domain and range
- The domain of [tex]\( f(x) = x^3 - 1 \)[/tex] is all real numbers because you can cube any real number.
- The range of [tex]\( f(x) = x^3 - 1 \)[/tex] is all real numbers because the cubic function can produce any real number output.
- The domain of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is all real numbers because you can take the cube root of any real number.
- The range of [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] is all real numbers because the cube root function can produce any real number output.
### Summary
(a) The function [tex]\( f(x) \)[/tex] is one-to-one, and the inverse function is given by:
[tex]\[ f^{-1}(x) = (x + 1)^{1/3} \][/tex]
(b) The graph of [tex]\( f(x) = x^3 - 1 \)[/tex] and [tex]\( f^{-1}(x) = (x + 1)^{1/3} \)[/tex] should be symmetric with respect to the line [tex]\( y = x \)[/tex].
(c) The domain and range of [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex] are:
- Domain of [tex]\( f(x) \)[/tex]: All real numbers
- Range of [tex]\( f(x) \)[/tex]: All real numbers
- Domain of [tex]\( f^{-1}(x) \)[/tex]: All real numbers
- Range of [tex]\( f^{-1}(x) \)[/tex]: All real numbers
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