f(x)=6x-4 g(x)=3x^(2)-2x-10 Write (g@f)(x) as an expression in terms of x. (g@f)(x)=

Write f(x): To find the composition of g(x) and f(x), denoted as (g@f)(x), we need to substitute the function f(x) into the function g(x) wherever there is an x in g(x).Write g(x): First, write down the function f(x): f(x) = 6x - 4.Substitute f into g: Next, write down the function g(x): g(x) = 3x^2 - 2x - 10.Expand squared term: Now, substitute f(x) into g(x) in place of x. This means wherever we see an x in g(x), we replace it with (6x - 4). (g@f)(x) = g(f(x)) = 3(6x - 4)^2 - 2(6x - 4) - 10.Distribute coefficients: Expand the square in the expression: (6x - 4)^2 = (6x - 4)(6x - 4) = 36x^2 - 48x + 16.Simplify expression: Substitute the expanded square back into the expression for (g@f)(x): (g@f)(x) = 3(36x^2 - 48x + 16) - 2(6x - 4) - 10.Combine like terms: Distribute the 3 and the -2 across the terms in the parentheses: (g@f)(x) = 3*36x^2 - 3*48x + 3*16 - 2*6x + 2*4 - 10.Final simplification: Simplify the expression by performing the multiplications: (g@f)(x) = 108x^2 - 144x + 48 - 12x + 8 - 10.Combine like terms in the expression: (g@f)(x) = 108x^2 - (144x + 12x) + (48 + 8 - 10).Finish simplifying by adding and subtracting the coefficients: (g@f)(x) = 108x^2 - 156x + 46.

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Math Problems

Algebra 1

Find the inverse of a linear function

Full solution

f(x)=6 x-4

\newline

g(x)=3 x^{2}-2 x-10

\newline

Write

(g \circ f)(x)

as an expression in terms of

x

\newline

(g \circ f)(x)=

Write $f(x)$ : To find the composition of $g(x)$ and $f(x)$ , denoted as $(g \circ f)(x)$ , we need to substitute the function $f(x)$ into the function $g(x)$ wherever there is an $x$ in $g(x)$ .
Write $g(x)$ : First, write down the function $f(x)$ : $f(x) = 6x - 4$ .
Substitute $f$ into $g$ : Next, write down the function $g(x)$ : $g(x) = 3x^2 - 2x - 10$ .
Expand squared term: Now, substitute $f(x)$ into $g(x)$ in place of $x$ . This means wherever we see an $x$ in $g(x)$ , we replace it with $(6x - 4)$ . $(g@f)(x) = g(f(x)) = 3(6x - 4)^2 - 2(6x - 4) - 10$ .
Distribute coefficients: Expand the square in the expression: \(6x - $4$ )^ $2$ = ( $6$ x - $4$ )( $6$ x - $4$ ) = $36$ x^ $2$ - $48$ x + $16$ \
Simplify expression: Substitute the expanded square back into the expression for $(g@f)(x)$ : $(g@f)(x) = 3(36x^2 - 48x + 16) - 2(6x - 4) - 10.$
Combine like terms: Distribute the $3$ and the $-2$ across the terms in the parentheses: $(g@f)(x) = 3 \times 36x^2 - 3 \times 48x + 3 \times 16 - 2 \times 6x + 2 \times 4 - 10$ .
Final simplification: Simplify the expression by performing the multiplications: $\newline$ $(g@f)(x) = 108x^2 - 144x + 48 - 12x + 8 - 10$ .
Final simplification: Simplify the expression by performing the multiplications: $\newline$ $(g@f)(x) = 108x^2 - 144x + 48 - 12x + 8 - 10$ .Combine like terms in the expression: $\newline$ $(g@f)(x) = 108x^2 - (144x + 12x) + (48 + 8 - 10)$ .
Final simplification: Simplify the expression by performing the multiplications: $\newline$ $(g@f)(x) = 108x^2 - 144x + 48 - 12x + 8 - 10$ .Combine like terms in the expression: $\newline$ $(g@f)(x) = 108x^2 - (144x + 12x) + (48 + 8 - 10)$ .Finish simplifying by adding and subtracting the coefficients: $\newline$ $(g@f)(x) = 108x^2 - 156x + 46$ .