Gateway exam practice - Math Insight

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Total points: 10

Time limit: 50 minutes

Solve for $r$. \begin{align*} - 3 X - 4 r - 10 = - 6 X - 2 r + 1 \end{align*} $r = $
Consider the function $f(t)=10(0.25)^t. \;\;$ 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.)
Solve the equation $10\left(z - 8\right) \left(z - 6\right)^{2} \left(z + 10\right) \left(4 z - 6\right) =0$.
The solutions are $z = $

If there are more than one solution, separate answers by commas
Compute the value of $f(f(\frac{1}{ 2 }))$ for the function $f(x)=4x(1-x)$.
$f(f(\frac{1}{ 2 })) =$
Simplify the inequality below: \begin{align*} -3<2 x + 3<5 \end{align*}

$\lt x \lt$
Find the equation for the line through the point $(-3,10)$ with slope given by $m=-3$
$y = $
Let $z(x) = - e^{x + 7}$ and $w(x) = - 5 x^{2} + x$. What is $z(w(x))$?
$z(w(x)) = $

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Hint
Use ^ to enter an exponent, e.g., for $a^b$, type a^b.
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Rewrite the exponential function \[ Q(t) = \frac{ 67 }{ 3^{t}4^{ t - 3 } } \] 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 = $
Factor the quadratic $z^{2} - 4 z - 21$.
Answer =
Let the variable $x$ be in the range \begin{align*} 3< x <9. \end{align*} If $y= 3 x + 4$, what is the range of the variable $y$?

$\lt y \lt$