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To determine which function has an inverse that is also a function, we need to analyze each function separately and check if their inverses satisfy the criteria of being a function. Only if each input of the inverse produces a unique output will the inverse be considered a function.
### Function [tex]\( g(x) = 2x - 3 \)[/tex]
1. To find the inverse, we set [tex]\( y = 2x - 3 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = 2x - 3 \][/tex]
2. Add 3 to both sides:
[tex]\[ y + 3 = 2x \][/tex]
3. Divide both sides by 2:
[tex]\[ x = \frac{y + 3}{2} \][/tex]
4. Thus, the inverse function is:
[tex]\[ g^{-1}(y) = \frac{y + 3}{2} \][/tex]
5. This inverse function produces a unique output for each input, thus [tex]\( g(x) = 2x - 3 \)[/tex] has an inverse that is also a function.
### Function [tex]\( k(x) = -9x^2 \)[/tex]
1. To find the inverse, we set [tex]\( y = -9x^2 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = -9x^2 \][/tex]
2. Divide both sides by -9:
[tex]\[ \frac{y}{-9} = x^2 \][/tex]
3. Taking the square root of both sides gives:
[tex]\[ x = \pm \sqrt{\frac{y}{-9}} \][/tex]
4. This means for each [tex]\( y \)[/tex], [tex]\( x \)[/tex] can be either [tex]\( \sqrt{\frac{y}{-9}} \)[/tex] or [tex]\(-\sqrt{\frac{y}{-9}} \)[/tex]. Since there are two possible values for each [tex]\( y \)[/tex], the inverse is not a function.
### Function [tex]\( f(x) = |x + 2| \)[/tex]
1. To find the inverse, we set [tex]\( y = |x + 2| \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = |x + 2| \][/tex]
2. This equation implies two cases:
[tex]\[ y = x + 2 \quad \text{or} \quad y = -(x + 2) \][/tex]
3. Solving these for [tex]\( x \)[/tex]:
[tex]\[ x = y - 2 \][/tex]
[tex]\[ x = -y - 2 \][/tex]
4. Since there are two possible values for each [tex]\( y \)[/tex], the inverse is not a function.
### Function [tex]\( w(x) = -20 \)[/tex]
1. This is a constant function, where [tex]\( w(x) = -20 \)[/tex] for all [tex]\( x \)[/tex].
2. A constant function does not have an inverse because it does not produce different output values for different input values.
3. Thus, [tex]\( w(x) = -20 \)[/tex] does not have an inverse function.
### Conclusion
Given the analyses above, the only function that has an inverse that is also a function is [tex]\( g(x) = 2x - 3 \)[/tex].
Thus, the function with an inverse that is also a function is:
[tex]\[ \boxed{g(x) = 2x - 3} \][/tex]
### Function [tex]\( g(x) = 2x - 3 \)[/tex]
1. To find the inverse, we set [tex]\( y = 2x - 3 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = 2x - 3 \][/tex]
2. Add 3 to both sides:
[tex]\[ y + 3 = 2x \][/tex]
3. Divide both sides by 2:
[tex]\[ x = \frac{y + 3}{2} \][/tex]
4. Thus, the inverse function is:
[tex]\[ g^{-1}(y) = \frac{y + 3}{2} \][/tex]
5. This inverse function produces a unique output for each input, thus [tex]\( g(x) = 2x - 3 \)[/tex] has an inverse that is also a function.
### Function [tex]\( k(x) = -9x^2 \)[/tex]
1. To find the inverse, we set [tex]\( y = -9x^2 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = -9x^2 \][/tex]
2. Divide both sides by -9:
[tex]\[ \frac{y}{-9} = x^2 \][/tex]
3. Taking the square root of both sides gives:
[tex]\[ x = \pm \sqrt{\frac{y}{-9}} \][/tex]
4. This means for each [tex]\( y \)[/tex], [tex]\( x \)[/tex] can be either [tex]\( \sqrt{\frac{y}{-9}} \)[/tex] or [tex]\(-\sqrt{\frac{y}{-9}} \)[/tex]. Since there are two possible values for each [tex]\( y \)[/tex], the inverse is not a function.
### Function [tex]\( f(x) = |x + 2| \)[/tex]
1. To find the inverse, we set [tex]\( y = |x + 2| \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ y = |x + 2| \][/tex]
2. This equation implies two cases:
[tex]\[ y = x + 2 \quad \text{or} \quad y = -(x + 2) \][/tex]
3. Solving these for [tex]\( x \)[/tex]:
[tex]\[ x = y - 2 \][/tex]
[tex]\[ x = -y - 2 \][/tex]
4. Since there are two possible values for each [tex]\( y \)[/tex], the inverse is not a function.
### Function [tex]\( w(x) = -20 \)[/tex]
1. This is a constant function, where [tex]\( w(x) = -20 \)[/tex] for all [tex]\( x \)[/tex].
2. A constant function does not have an inverse because it does not produce different output values for different input values.
3. Thus, [tex]\( w(x) = -20 \)[/tex] does not have an inverse function.
### Conclusion
Given the analyses above, the only function that has an inverse that is also a function is [tex]\( g(x) = 2x - 3 \)[/tex].
Thus, the function with an inverse that is also a function is:
[tex]\[ \boxed{g(x) = 2x - 3} \][/tex]
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