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2 years ago

What are the domain, range, and asymptote of h(x) = 6x – 4?

1 answer:

7 0

h(x)=6x−4 or h(x)=6x−4
If the former, the domain is all real numbers, often denoted by the symbol R. 
the range is (0,∞)
asymptote is y = 0

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3 years ago

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Looks like the function on the right hand side is

$f(t)=\begin{cases}1&\text{for }t$

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$h(t-a)=\begin{cases}1&\text{for }t\ge a\\0&\text{for }t>a\end{cases}$

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Now for the ODE: take the Laplace transform of both sides:

$y'(t)+y(t)=f(t)$

$\implies s Y(s)-y(0)+Y(s)=\dfrac{1-2e^{-4s}}s$

Solve for Y(s), then take the inverse transform to solve for y(t):

$(s+1)Y(s)=\dfrac{1-e^{-4s}}s$

$Y(s)=\dfrac{1-e^{-4s}}{s(s+1)}$

$Y(s)=(1-e^{-4s})\left(\dfrac1s-\dfrac1{s+1}\right)$

$Y(s)=\dfrac1s-\dfrac{e^{-4s}}s-\dfrac1{s+1}+\dfrac{e^{-4s}}{s+1}$

$\implies y(t)=1-h(t-4)-e^{-t}+e^{-(t-4)}h(t-4)$

$\boxed{y(t)=1-e^{-t}-h(t-4)(1-e^{-(t-4)})}$

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2 years ago