smc-6640-50-\(f(x)+g(x)\)

Functions, EXT1 F1 2022 HSC 4 MC

The diagram shows the graph of the sum of the functions `f(x)` and `g(x)`.

Which of the following best represents the graphs of both `f(x)` and `g(x)`?

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`A`

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`text{By Elimination,}`

`text{Consider the}\ ytext{-axis intercept of both graphs in each option:}`

`B and C\ text{will have a positive “net”}\ ytext{-value intercept (Eliminate}\ B\ \text{and}\ C).`

`y=f(x)+g(x)=0\ \ text{when the two graphs are equidistant from the}`

`xtext{-axis (Eliminate}\ D).`

`=>A`

Which curve best represents the graph of the function `f(x) = -a sin x + b cos x` given that the constants `a` and `b` are both positive?

A.		B.
C.		D.

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`D`

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`text(By elimination:)`

`text(At)\ \ x = 0, -asinx = 0\ \ text(and)\ \ bcosx = b`

COMMENT: When told constants are “positive”, pay close attention!

`:. f(0) = b > 0`

`->\ text(Eliminate A and C)`

`f^{′}(x) = -acosx-bsinx`

`text(At)\ \ x = 0, -acos x = -a\ \ text(and)\ \ bsinx = 0`

`:. f′(0) = -a < 0`

`->\ text(Eliminate B)`

`=>\ D`

The diagram shows the graph of the function `f(x) = x/(x-1)`.

If `g(x)=x`, draw the graph of `y = f(x) + g(x)` (2 marks)

--- 12 WORK AREA LINES (style=lined) ---

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`text(As)\ x -> ∞, \ y -> x + 1`

`text(As)\ x ->-∞, \ y -> x + 1`