Get detailed and reliable answers to your questions on IDNLearn.com. Get accurate and comprehensive answers from our network of experienced professionals.
Sagot :
To find the inverse of the function [tex]\( f(x) = 2x + 3 \)[/tex], follow these steps:
1. Write the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 2y + 3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ x - 3 = 2y \][/tex]
[tex]\[ y = \frac{x - 3}{2} \][/tex]
4. Rewrite [tex]\( y \)[/tex] as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x - 3}{2} \][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \frac{x - 3}{2} \)[/tex].
Now, let's compare this with the given options:
- Option 1: [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 2: [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 3: [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
- Option 4: [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]
Clearly, the correct option that matches our derived inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2}(x - 3) \][/tex]
Simplifying further,
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
Therefore, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
1. Write the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 2y + 3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ x - 3 = 2y \][/tex]
[tex]\[ y = \frac{x - 3}{2} \][/tex]
4. Rewrite [tex]\( y \)[/tex] as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x - 3}{2} \][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \frac{x - 3}{2} \)[/tex].
Now, let's compare this with the given options:
- Option 1: [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 2: [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 3: [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
- Option 4: [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]
Clearly, the correct option that matches our derived inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2}(x - 3) \][/tex]
Simplifying further,
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
Therefore, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.