Question

1. what is $f(g(x))$?

$f(x) = 3x + 5$ and $g(x) = 5x - 1$
$\bigcirc$ $f(g(x)) = 15x + 24$
$\bigcirc$ $f(g(x)) = 15x + 2$
$\bigcirc$ $f(g(x)) = 15x - 1$
$\bigcirc$ $f(g(x)) = 15x + 5$

1. multiply the matrices.

$1, 2, 3 \

$$\begin{bmatrix} 3 \\\\ 1 \\\\ -1 \\end{bmatrix}$$

$
$\bigcirc$ $12$
$\bigcirc$ $2$
$\bigcirc$ $2$
$\bigcirc$ $\

$$\begin{bmatrix} 3 & 3 & 3 \\\\ 3 & 3 & 3 \\\\ 3 & 3 & 3 \\end{bmatrix}$$

$

1. use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots of the quadratic equation.

$y = x^2 + x + 1$
$\bigcirc$ $x = -\frac{1}{2} \pm i\frac{\sqrt{5}}{2}$
$\bigcirc$ $x = -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}$
$\bigcirc$ $x = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}$
$\bigcirc$ $x = -\frac{1}{2} \pm \frac{\sqrt{5}}{2}$

Explanation:

Question 10

Step1: Substitute $g(x)$ into $f(x)$

$f(g(x)) = 3(5x-1) + 5$

Step2: Expand and simplify

$f(g(x)) = 15x - 3 + 5 = 15x + 2$

Step1: Compute matrix dot product

$(1\times3) + (2\times1) + (3\times(-1))$

Step2: Calculate the sum

$3 + 2 - 3 = 2$

Step1: Identify $a,b,c$

For $y=x^2+x+1$, $a=1$, $b=1$, $c=1$

Step2: Substitute into quadratic formula

$x = \frac{-1\pm\sqrt{1^2-4(1)(1)}}{2(1)}$

Step3: Simplify discriminant

$\sqrt{1-4} = \sqrt{-3} = i\sqrt{3}$

Step4: Final simplification

$x = -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}$

Answer:

$f(g(x)) = 15x + 2$

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Question 11