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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)=

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

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Evaluate an exponential function

Full solution

Q.

g(x)=3-x^{2}

\newline

h(x)=4-3 x

\newline

Write

(g \circ h)(x)

as an expression in terms of

x

\newline

(g \circ h)(x)=

Substitute Functions: To find the composition of functions $(g@h)(x)$ , we need to substitute the entire function $h(x)$ into $g(x)$ for every instance of $x$ . The function $g(x)$ is defined as $g(x) = 3 - x^2$ . The function $h(x)$ is defined as $h(x) = 4 - 3x$ . So, we will replace every $x$ in $g(x)$ with the expression $h(x)$ $0$ .
Square Expression: Substitute $h(x)$ into $g(x)$ : $g(h(x)) = 3 - (h(x))^2$ . Now, replace $h(x)$ with its expression: $g(h(x)) = 3 - (4 - 3x)^2$ .
Expand Expression: Now we need to square the expression $(4 - 3x)$ : $(4 - 3x)^2 = (4 - 3x) \times (4 - 3x)$ . We will use the FOIL method (First, Outer, Inner, Last) to expand this expression.
Simplify Result: Expanding $(4 - 3x) * (4 - 3x)$ gives us: First: $4*4 = 16$ , Outer: $4*(-3x) = -12x$ , Inner: $(-3x)*4 = -12x$ , Last: $(-3x)*(-3x) = 9x^2$ . Adding these together, we get: $16 - 12x - 12x + 9x^2$ .
Substitute Back: Simplify the expression: $16 - 12x - 12x + 9x^2$ becomes $16 - 24x + 9x^2$ . Now we have the squared term for the composition of functions.
Distribute Negative: Substitute the simplified squared term back into the expression for $g(h(x))$ : $g(h(x)) = 3 - (16 - 24x + 9x^2)$ .
Combine Like Terms: Distribute the negative sign through the parentheses: $g(h(x)) = 3 - 16 + 24x - 9x^2$ .
Final Expression: Combine like terms: $g(h(x)) = -13 + 24x - 9x^2$ . This is the final expression for $(g@h)(x)$ in terms of $x$ .

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