Gateway exam practice - Math Insight

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Section:
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Total points: 10

Time limit: 50 minutes

Let the variable $x$ be in the range \begin{align*} 0< x <7. \end{align*} If $y= 7 x + 3$, what is the range of the variable $y$?

$\lt y \lt$
Simplify the inequality below: \begin{align*} \left|\frac{3 x}{5} + 1\right|<5 \end{align*}

$\lt x \lt$
Rewrite the exponential function \[ Q(t) = \frac{ 7 }{ 5^{t}4^{ t - 4 } } \] in the form $Q(t)=ab^t$, where we call $a$ the “initial value” (the value when $t=0$) and $b$ the “growth factor.” In this form:
$a = $

$b = $
Consider the function $f(t)=13(0.4)^t. \;\;$ Find the half-life, i.e. find $t$ such that $f(t)=\frac{1}{2}f(0).$
Half-life =

(If you round your answer, include at least 4 significant digits.)
Let $w(x) = - 4 x^{2} + 2$ and $z(x) = \sqrt{ 7 x}$. What is $w(z(x))$?
$w(z(x)) = $
Solve for $q$. \begin{align*} - 6 q + 10 z = 6 q - 2 z - 4 \end{align*} $q = $
Compute the value of $f(f(\frac{1}{ 4 }))$ for the function $f(x)=2x(1-x)$.
$f(f(\frac{1}{ 4 })) =$
Find the equation for the line through the point $(-10,9)$ with slope given by $m=- \frac{13}{7}$
$y = $
Factor the quadratic $z^{2} + 14 z + 45$.
Answer =
Solve the equation $9\left(y - 10\right) \left(y - 9\right)^{2} \left(y - 6\right)^{2} \left(7 y + 8\right) =0$.
The solutions are $y = $

If there are more than one solution, separate answers by commas