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Sagot :
To find the inverse of the function [tex]\( g(x) = 8x + 3 \)[/tex], you can follow these steps:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 8x + 3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to set up for inverting the function:
[tex]\[ x = 8y + 3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], first subtract the constant term from both sides:
[tex]\[ x - 3 = 8y \][/tex]
Then, divide both sides by the coefficient of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x - 3}{8} \][/tex]
4. Simplify the expression:
We can further simplify the above expression:
[tex]\[ y = \frac{x}{8} - \frac{3}{8} \][/tex]
5. Rewrite the inverse function:
The inverse function [tex]\( g^{-1}(x) \)[/tex] is written in terms of [tex]\( x \)[/tex]:
[tex]\[ g^{-1}(x) = \frac{x}{8} - \frac{3}{8} \][/tex]
Therefore, the simplified form of the inverse function is:
[tex]\[ g^{-1}(x) = \frac{x}{8} - \frac{3}{8} \][/tex]
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 8x + 3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to set up for inverting the function:
[tex]\[ x = 8y + 3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], first subtract the constant term from both sides:
[tex]\[ x - 3 = 8y \][/tex]
Then, divide both sides by the coefficient of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x - 3}{8} \][/tex]
4. Simplify the expression:
We can further simplify the above expression:
[tex]\[ y = \frac{x}{8} - \frac{3}{8} \][/tex]
5. Rewrite the inverse function:
The inverse function [tex]\( g^{-1}(x) \)[/tex] is written in terms of [tex]\( x \)[/tex]:
[tex]\[ g^{-1}(x) = \frac{x}{8} - \frac{3}{8} \][/tex]
Therefore, the simplified form of the inverse function is:
[tex]\[ g^{-1}(x) = \frac{x}{8} - \frac{3}{8} \][/tex]
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