Question

How do you graph, find the intercepts and state the domain and range of `f(x)=6^x+3`?

Answer 1

y-intercept `->(x,y)=(0,4)`

No x-intercept

domain `color(white)(.)->{x: x in (-oo,+oo)}`

range `" "->{y: y in (+3,+oo)}`

Explanation:

Set `" "y=6^x+3`

`color(blue)("Determine the y-intercept")`

Set `x=0` giving

`y=6^0+3 = 4`
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`color(blue)("Determine the x-intercept")`

Set `y=0=6^x+3`

`6^x=-3`

Take logs of both sides

`xln(6)=ln(-3)`

But ln(-3) is undefined thus there is no x-intercept
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`color(blue)("Determine end behaviour "x->-oo)`

`lim_(x->-oo) 6^x+3 = lim_(z->+oo) 1/6^z+3 ->k = 3`
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`color(blue)("Determine end behaviour "x->+oo)`

`lim_(x->+oo) 6^x+3" " ->" "k+3 =oo+3=oo`
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