Find the domain and range for the indicated function.
-If $f ( x ) = 4 x ^ { 2 } + 6 x + 8$ and $g ( x ) = 6 x - 7$, find $g ( f ( x ) )$
A) $4 x ^ { 2 } + 36 x + 41$
B) $24 x ^ { 2 } + 36 x + 55$
C) $4 x ^ { 2 } + 6 x + 1$
D) $24 x ^ { 2 } + 36 x + 41$

Question

Quizplus · Accepted Answer

We need to first plug in $f(x)$ into $g(x)$, so we get:

$$g(f(x)) = 6f(x)-7$$

Now we can substitute $f(x)$ in:

$$g(f(x)) = 6(4 x ^ { 2 } + 6 x + 8)-7$$

$$g(f(x)) = 24 x ^ { 2 } + 36 x + 41$$

Therefore, the answer is D.

As for the domain and range of the composed function, since $f(x)$ is a polynomial and $g(x)$ is a linear function, the domain of the composed function is all real numbers.

For the range, we notice that the quadratic term in $f(x)$ dominates as $x \to \pm \infty$, so the range of $f(x)$ is bounded below, but not above. Therefore, the range of the composed function $g(f(x))$ is all real numbers greater than or equal to some lower bound. Specifically, since $4x^2 + 6x + 8$ is always nonnegative (it's a quadratic with a positive leading coefficient), the lowest possible value of $g(f(x))$ is achieved at $x=-\frac{3}{4}$, where $f(x)$ is equal to 6, so the range of the composed function is $[6, \infty)$.

Find the Domain and Range for the Indicated Function f(x)=4x2+6x+8f ( x ) = 4 x ^ { 2 } + 6 x + 8f(x)=4x2+6x+8

Find the Domain and Range for the Indicated Function $f ( x ) = 4 x ^ { 2 } + 6 x + 8$