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Mathematics

1 answer:

KIM [24]3 years ago

8 0

Answer:

k = - 2

Step-by-step explanation:

Lines g(x) and f(x) passes through the points (2, 0), (0, 2) & (4, 0), (0, 4)

Since both the lines are parallel, so slopes of lines g(x) and f(x) would be same.

Therefore,

Slope of line g = Slope of line f = (2-0)/(0 - 2) = 2/-2 = - 1

Equation of line g(x)

For point (0, 2)

y- intercept (b) = 2

y = mx + b

g(x) = - 1x + 2 [y = g(x)]

g(x) = - x + 2

Equation of line f(x)

For point (0, 4)

y- intercept (b) = 4

y = mx + b

f(x) = - 1x + 4 [y = g(x)]

f(x) = - x + 4

It is given that:

g(x) = f(x) + k

g(x) - f(x) = k

(-x + 2) - (-x + 4) = k

-x + 2 + x - 4 = k

2 - 4 = k

-2 = k

k = - 2

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I want to make sure i got this right. PLS HELP ASAP I NEED IT CALC 1 one question

IrinaK [193]

$\bf f(x)=
\begin{cases}
x^3&x\le 1\\
mx+b&x\ \textgreater \ 1
\end{cases}$

now, a function may well be continuous, but it may not be differentiable.

graph wise, differentiability means, a smooth curve, with no abrupt spikes or drops or cuts.

so for this piece-wise to be differentiable, the subfunctions must meet at junctions smoothly, with no abrupt changes, so mx + b, must blend in with x³ smoothly.

now, that can only occur if mx + b has the same slope as x³ at that point where they meet. The point where they meet is at x = 1, well, let's check what is the slope of x³ at x = 1.

$\bf \cfrac{d}{dx}[x^3]\implies \left. 3x^2\cfrac{}{}\right|_{x=1}\implies 3(1)^2\implies 3$

therefore "m" for mx+b, must be 3 then.

now, what's the value of x³ at x = 1? well is just (1)³, which is 1.

so for 3x + b, when x = 1, it must also equals 1 as well, that way it meets x³, thus

3x + b

3(1) + b = 1

b = -2

check the graph below.