### Question 03: Find the Inverse Function of [tex]\( j(x) = \frac{2 - x}{x} \)[/tex]
To find the inverse function of [tex]\( j(x) \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( j(x) = y \)[/tex].
1. Start with the equation:
[tex]\[ y = \frac{2 - x}{x} \][/tex]
2. Multiply both sides by [tex]\( x \)[/tex] to eliminate the denominator:
[tex]\[ yx = 2 - x \][/tex]
3. Rearrange the equation to isolate terms involving [tex]\( x \)[/tex]:
[tex]\[ yx + x = 2 \][/tex]
[tex]\[ x(y + 1) = 2 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2}{y + 1} \][/tex]
Thus, the inverse function [tex]\( j^{-1}(y) \)[/tex] is:
[tex]\[ j^{-1}(y) = \frac{2}{y + 1} \][/tex]
### Question 04: Find the Value of [tex]\( g(x) = \frac{9 - 9x^2}{x^2} \)[/tex] if [tex]\( x = g^{-1}(7) \)[/tex]
To find [tex]\( g^{-1}(7) \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( g(x) = 7 \)[/tex].
1. Start with the equation:
[tex]\[ 7 = \frac{9 - 9x^2}{x^2} \][/tex]
2. Separate the terms in the fraction:
[tex]\[ 7 = \frac{9}{x^2} - 9 \][/tex]
3. Add 9 to both sides to isolate the fraction:
[tex]\[ 16 = \frac{9}{x^2} \][/tex]
4. Solve for [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = \frac{9}{16} \][/tex]
[tex]\[ x = \pm \frac{3}{4} \][/tex]
Now we have the possible values for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3}{4} \text{ or } x = -\frac{3}{4} \][/tex]
Choosing [tex]\( x = -\frac{3}{4} \)[/tex] (as obtained), we substitute back into [tex]\( g(x) \)[/tex] to find the value of [tex]\( g(x) \)[/tex] when [tex]\( x = g^{-1}(7) \)[/tex]:
1. Substitute [tex]\( x = -\frac{3}{4} \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g\left( -\frac{3}{4} \right) = \frac{9 - 9 \left( -\frac{3}{4} \right)^2}{\left( -\frac{3}{4} \right)^2} \][/tex]
2. Simplify the equation:
[tex]\[ g\left( -\frac{3}{4} \right) = \frac{9 - 9 \left(\frac{9}{16}\right)}{\frac{9}{16}} \][/tex]
[tex]\[ g\left( -\frac{3}{4} \right) = \frac{9 - \frac{81}{16}}{\frac{9}{16}} \][/tex]
[tex]\[ g\left( -\frac{3}{4} \right) = \frac{\frac{144 - 81}{16}}{\frac{9}{16}} \][/tex]
[tex]\[ g\left( -\frac{3}{4} \right) = \frac{\frac{63}{16}}{\frac{9}{16}} \][/tex]
[tex]\[ g\left( -\frac{3}{4} \right) = 7 \][/tex]
Therefore, the value of [tex]\( g(x) \)[/tex] when [tex]\( x = g^{-1}(7) \)[/tex] is:
[tex]\[ g\left( -\frac{3}{4} \right) = 7 \][/tex]
So, summarizing the results:
- The inverse function of [tex]\( j(x) = \frac{2 - x}{x} \)[/tex] is [tex]\( j^{-1}(y) = \frac{2}{y + 1} \)[/tex].
- The value of [tex]\( g(x) = \frac{9 - 9 x^2}{x^2} \)[/tex] if [tex]\( x = g^{-1}(7) \)[/tex] is [tex]\( 7 \)[/tex].
