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${:[g(x)=3-x^(2)],[h(x)=4-3x]:} Write (g@h)(x) as an expression in terms of x. (g@h)(x)=$

$\begin{array}{l} g(x)=3-x^{2} \\ h(x)=4-3 x \end{array}$ $\newline$ Write $(g \circ h)(x)$ as an expression in terms of $x$ . $\newline$ $(g \circ h)(x)=$

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Compare linear and exponential growth

Full solution

Q.

\begin{array}{l} g(x)=3-x^{2} \\ h(x)=4-3 x \end{array}

\newline

Write

(g \circ h)(x)

as an expression in terms of

x

.

\newline

(g \circ h)(x)=

Find composition of functions: We need to find the composition of the functions $g(x)$ and $h(x)$ , which is written as $(g@h)(x)$ . This means we need to substitute the function $h(x)$ into the function $g(x)$ .
Write down $g(x)$ and $h(x)$ : First, let's write down the functions $g(x)$ and $h(x)$ for clarity: $\newline$ $g(x) = 3 - x^2$ $\newline$ $h(x) = 4 - 3x$ $\newline$ Now, to find $(g@h)(x)$ , we replace every instance of $x$ in $g(x)$ with $h(x)$ .
Perform substitution: Perform the substitution: $\newline$ $(g@h)(x) = g(h(x)) = 3 - (h(x))^2$ $\newline$ Now, we need to square the function $h(x)$ .
Square $h(x)$ : Square $h(x)$ : $(h(x))^2 = (4 - 3x)^2$ To square a binomial, we use the formula $(a - b)^2 = a^2 - 2ab + b^2$ .
Apply formula to square $h(x)$ : Apply the formula to square $h(x)$ : $(4 - 3x)^2 = 4^2 - 2\times(4)\times(3x) + (3x)^2$ Calculate the square and products: $(4 - 3x)^2 = 16 - 24x + 9x^2$
Calculate square and products: Now, substitute $(4 - 3x)^2$ back into the expression for $(g@h)(x)$ : $\newline$ $(g@h)(x) = 3 - (16 - 24x + 9x^2)$ $\newline$ Simplify the expression by distributing the negative sign: $\newline$ $(g@h)(x) = 3 - 16 + 24x - 9x^2$
Substitute back into $(g@h)(x)$ : Combine like terms: $\newline$ $(g@h)(x) = -13 + 24x - 9x^2$ $\newline$ Reorder the terms to write the quadratic in standard form: $\newline$ $(g@h)(x) = -9x^2 + 24x - 13$

More problems from Compare linear and exponential growth

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days to pick up the motorcycle.

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\newline

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g(t)=9^t

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Both of these functions grow as

x

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\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 8 months ago

Question

Both of these functions grow as

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gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 8 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 8 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

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Posted 9 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

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v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 9 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 9 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

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o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 9 months ago

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