To solve for [tex]\( m \)[/tex] given the functions [tex]\( f(a) = \frac{a}{m} \)[/tex], [tex]\( g(x) = 5x - 7 \)[/tex], and the composite function [tex]\( f[g(x)] = 15x - 21 \)[/tex], follow these steps:
1. Calculate [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 5x - 7
\][/tex]
2. Compose the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
[tex]\[
f[g(x)] = f(5x - 7)
\][/tex]
3. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(a) \)[/tex]:
[tex]\[
f(5x - 7) = \frac{5x - 7}{m}
\][/tex]
4. We know that the composition [tex]\( f[g(x)] \)[/tex] is given as:
[tex]\[
f[g(x)] = 15x - 21
\][/tex]
5. Set the composed function equal to the given expression:
[tex]\[
\frac{5x - 7}{m} = 15x - 21
\][/tex]
6. To find [tex]\( m \)[/tex], equate the coefficients of corresponding terms on both sides of the equation.
- Coefficient of [tex]\( x \)[/tex]:
[tex]\[
\frac{5}{m} = 15
\][/tex]
- Constant term:
[tex]\[
\frac{-7}{m} = -21
\][/tex]
7. Solve for [tex]\( m \)[/tex] from the coefficient of [tex]\( x \)[/tex]:
[tex]\[
\frac{5}{m} = 15 \implies m = \frac{5}{15} = \frac{1}{3}
\][/tex]
8. Solve for [tex]\( m \)[/tex] from the constant term:
[tex]\[
\frac{-7}{m} = -21 \implies m = \frac{-7}{-21} = \frac{1}{3}
\][/tex]
9. Both calculations give the same value of [tex]\( m \)[/tex].
Thus, the value of [tex]\( m \)[/tex] is [tex]\( \boxed{\frac{1}{3}} \)[/tex].
