Gateway exam practice - Math Insight

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Total points: 10

Time limit: 50 minutes

Consider the function $f(t)=10(0.5)^t. \;\;$ 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 $w(x) = x^{2} + 7$ and $u(x) = \sqrt{ 2 x}$. What is $w(u(x))$?
$w(u(x)) = $
Simplify the inequality below: \begin{align*} \left|\frac{7 x}{2} + 6\right|<9 \end{align*}

$\lt x \lt$
The difference between two positive numbers is 1 and the sum of their squares is 145.
The numbers are

Separate answers by a comma.
Solve for U. \begin{align*} 6 = - 7 U \end{align*} $U = $
Compute the value of $f(f(\frac{1}{ 2 }))$ for the function $f(x)=2x(1-x)$.
$f(f(\frac{1}{ 2 })) =$
Solve the equation $-4\left(y - 5\right)^{2} \left(y - 2\right) \left(y + 9\right)^{2} \left(9 y + 10\right) =0$.
The solutions are $y = $

If there are more than one solution, separate answers by commas
Find the equation for the line through the points $(-1,1)$ and $(-4,-9)$.
$y = $
Let the variable $x$ be in the range \begin{align*} 2< x <7. \end{align*} If $y= 6 x + 2$, what is the range of the variable $y$?

$\lt y \lt$
Write the function $f(x)=9e^{ 6x+7 }$ in the form $f(x)=Ae^{kx}$. What are the values of the parameters $A$ and $k$?
$A=$
, $k=$

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Hint
Use ^ to enter an exponent, e.g., for $a^b$, type a^b.
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