To determine which equation could represent the function [tex]\( m \)[/tex] given that [tex]\( m(4) = 9 \)[/tex] and [tex]\( m \)[/tex] is nonlinear, we need to evaluate each option at [tex]\( x = 4 \)[/tex] and see which one equals 9.
Let's check each option step-by-step:
Option A: [tex]\( m(x) = x^2 - 7 \)[/tex]
Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ m(4) = 4^2 - 7 \][/tex]
[tex]\[ m(4) = 16 - 7 \][/tex]
[tex]\[ m(4) = 9 \][/tex]
This option satisfies the condition [tex]\( m(4) = 9 \)[/tex].
Option B: [tex]\( m(x) = 9 \)[/tex]
Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ m(4) = 9 \][/tex]
This option also satisfies the condition [tex]\( m(4) = 9 \)[/tex]. However, the question specifies that the function is nonlinear, and [tex]\( m(x) = 9 \)[/tex] represents a constant (linear) function. Hence, this option is not viable despite satisfying the condition [tex]\( m(4) = 9 \)[/tex].
Option C: [tex]\( m(x) = \sqrt{x} + 1 \)[/tex]
Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ m(4) = \sqrt{4} + 1 \][/tex]
[tex]\[ m(4) = 2 + 1 \][/tex]
[tex]\[ m(4) = 3 \][/tex]
This option does not satisfy the condition [tex]\( m(4) = 9 \)[/tex].
Option D: [tex]\( m(x) = \frac{2}{3} x - 2 \)[/tex]
Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[ m(4) = \frac{2}{3} \cdot 4 - 2 \][/tex]
[tex]\[ m(4) = \frac{8}{3} - 2 \][/tex]
[tex]\[ m(4) = \frac{8}{3} - \frac{6}{3} \][/tex]
[tex]\[ m(4) = \frac{2}{3} \][/tex]
This option does not satisfy the condition [tex]\( m(4) = 9 \)[/tex].
Considering all these evaluations, the only option that satisfies the condition [tex]\( m(4) = 9 \)[/tex] and represents a nonlinear function is:
Option A: [tex]\( m(x) = x^2 - 7 \)[/tex]
