Get the answers you need from a community of experts on IDNLearn.com. Get accurate and comprehensive answers to your questions from our community of knowledgeable 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 value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.