g(x)=x^2-1 f(x)=4x+4 Fin d (gcirc f)(x)

To find the composition of the functions \( g(x) \) and \( f(x) \), denoted as \( (g \circ f)(x) \), we need to substitute \( f(x) \) into \( g(x) \).<br /><br />Given:<br />\[ g(x) = x^2 - 1 \]<br />\[ f(x) = 4x + 4 \]<br /><br />First, we find \( f(x) \):<br />\[ f(x) = 4x + 4 \]<br /><br />Next, we substitute \( f(x) \) into \( g(x) \):<br />\[ (g \circ f)(x) = g(f(x)) = g(4x + 4) \]<br /><br />Now, we substitute \( 4x + 4 \) into \( g(x) \):<br />\[ g(4x + 4) = (4x + 4)^2 - 1 \]<br /><br />We expand \( (4x + 4)^2 \):<br />\[ (4x + 4)^2 = (4x + 4)(4x + 4) = 16x^2 + 32x + 16 \]<br /><br />So,<br />\[ g(4x + 4) = 16x^2 + 32x + 16 - 1 \]<br />\[ g(4x + 4) = 16x^2 + 32x + 15 \]<br /><br />Therefore, the composition \( (g \circ f)(x) \) is:<br />\[ (g \circ f)(x) = 16x^2 + 32x + 15 \]