Math / Algebra

QuestionGiven that $\mathrm{g}(\mathrm{x})=3 \mathrm{x}^{2}-4 \mathrm{x}+2$ , find each of the following. a) $g(0)$ b) $g(-2)$ c) $g(3)$ d) $g(-x)$ e) $g(1-t)$

Studdy Solution

STEP 1

1. The function $\mathrm{g}(\mathrm{x})=3 \mathrm{x}^{2}-4 \mathrm{x}+2$ is a quadratic polynomial.
2. To find $\mathrm{g}(\mathrm{x})$ at specific values, we substitute the given value into the function and simplify.
3. For part (d) and (e), we substitute $-x$ and $1-t$ respectively into the function and simplify.

STEP 2

1. Evaluate $g(0)$ by substituting $0$ into the function.
2. Evaluate $g(-2)$ by substituting $-2$ into the function.
3. Evaluate $g(3)$ by substituting $3$ into the function.
4. Evaluate $g(-x)$ by substituting $-x$ into the function.
5. Evaluate $g(1-t)$ by substituting $1-t$ into the function.

STEP 3

Evaluate $g(0)$ .
$g(0) = 3(0)^2 - 4(0) + 2 = 2$

STEP 4

Evaluate $g(-2)$ .
$g(-2) = 3(-2)^2 - 4(-2) + 2$

STEP 5

Simplify the expression for $g(-2)$ .
$g(-2) = 3(4) + 8 + 2 = 12 + 8 + 2 = 22$

STEP 6

Evaluate $g(3)$ .
$g(3) = 3(3)^2 - 4(3) + 2$

STEP 7

Simplify the expression for $g(3)$ .
$g(3) = 3(9) - 12 + 2 = 27 - 12 + 2 = 17$

STEP 8

Evaluate $g(-x)$ by substituting $-x$ into the function $g(x)$ .
$g(-x) = 3(-x)^2 - 4(-x) + 2$

STEP 9

Simplify the expression for $g(-x)$ .
$g(-x) = 3x^2 + 4x + 2$

STEP 10

Evaluate $g(1-t)$ by substituting $1-t$ into the function $g(x)$ .
$g(1-t) = 3(1-t)^2 - 4(1-t) + 2$

STEP 11

Expand and simplify the expression for $g(1-t)$ .
$g(1-t) = 3(1 - 2t + t^2) - 4(1 - t) + 2$ $= 3 - 6t + 3t^2 - 4 + 4t + 2$ $= 3t^2 - 2t + 1$
The solutions are: a) $g(0) = 2$ b) $g(-2) = 22$ c) $g(3) = 17$ d) $g(-x) = 3x^2 + 4x + 2$ e) $g(1-t) = 3t^2 - 2t + 1$

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QuestionGiven that $\mathrm{g}(\mathrm{x})=3 \mathrm{x}^{2}-4 \mathrm{x}+2$ , find each of the following. a) $g(0)$ b) $g(-2)$ c) $g(3)$ d) $g(-x)$ e) $g(1-t)$

Studdy Solution

STEP 1

STEP 2

1. Evaluate $g(0)$ by substituting $0$ into the function.
2. Evaluate $g(-2)$ by substituting $-2$ into the function.
3. Evaluate $g(3)$ by substituting $3$ into the function.
4. Evaluate $g(-x)$ by substituting $-x$ into the function.
5. Evaluate $g(1-t)$ by substituting $1-t$ into the function.

STEP 3

Evaluate $g(0)$ .
$g(0) = 3(0)^2 - 4(0) + 2 = 2$

STEP 4

Evaluate $g(-2)$ .
$g(-2) = 3(-2)^2 - 4(-2) + 2$

STEP 5

Simplify the expression for $g(-2)$ .
$g(-2) = 3(4) + 8 + 2 = 12 + 8 + 2 = 22$

STEP 6

Evaluate $g(3)$ .
$g(3) = 3(3)^2 - 4(3) + 2$

STEP 7

Simplify the expression for $g(3)$ .
$g(3) = 3(9) - 12 + 2 = 27 - 12 + 2 = 17$

STEP 8

Evaluate $g(-x)$ by substituting $-x$ into the function $g(x)$ .
$g(-x) = 3(-x)^2 - 4(-x) + 2$

STEP 9

Simplify the expression for $g(-x)$ .
$g(-x) = 3x^2 + 4x + 2$

STEP 10

Evaluate $g(1-t)$ by substituting $1-t$ into the function $g(x)$ .
$g(1-t) = 3(1-t)^2 - 4(1-t) + 2$

STEP 11

Expand and simplify the expression for $g(1-t)$ .
$g(1-t) = 3(1 - 2t + t^2) - 4(1 - t) + 2$ $= 3 - 6t + 3t^2 - 4 + 4t + 2$ $= 3t^2 - 2t + 1$
The solutions are: a) $g(0) = 2$ b) $g(-2) = 22$ c) $g(3) = 17$ d) $g(-x) = 3x^2 + 4x + 2$ e) $g(1-t) = 3t^2 - 2t + 1$

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QuestionGiven that g(x)=3x2−4x+2\mathrm{g}(\mathrm{x})=3 \mathrm{x}^{2}-4 \mathrm{x}+2g(x)=3x2−4x+2, find each of the following. a) g(0)g(0)g(0) b) g(−2)g(-2)g(−2) c) g(3)g(3)g(3) d) g(−x)g(-x)g(−x) e) g(1−t)g(1-t)g(1−t)

Studdy Solution

STEP 1

STEP 2

STEP 3

Evaluate g(0)g(0)g(0).g(0)=3(0)2−4(0)+2=2 g(0) = 3(0)^2 - 4(0) + 2 = 2 g(0)=3(0)2−4(0)+2=2

STEP 4

Evaluate g(−2)g(-2)g(−2).g(−2)=3(−2)2−4(−2)+2 g(-2) = 3(-2)^2 - 4(-2) + 2 g(−2)=3(−2)2−4(−2)+2

STEP 5

Simplify the expression for g(−2)g(-2)g(−2).g(−2)=3(4)+8+2=12+8+2=22 g(-2) = 3(4) + 8 + 2 = 12 + 8 + 2 = 22 g(−2)=3(4)+8+2=12+8+2=22

STEP 6

Evaluate g(3)g(3)g(3).g(3)=3(3)2−4(3)+2 g(3) = 3(3)^2 - 4(3) + 2 g(3)=3(3)2−4(3)+2

STEP 7

Simplify the expression for g(3)g(3)g(3).g(3)=3(9)−12+2=27−12+2=17 g(3) = 3(9) - 12 + 2 = 27 - 12 + 2 = 17 g(3)=3(9)−12+2=27−12+2=17

STEP 8

Evaluate g(−x)g(-x)g(−x) by substituting −x-x−x into the function g(x)g(x)g(x).g(−x)=3(−x)2−4(−x)+2 g(-x) = 3(-x)^2 - 4(-x) + 2 g(−x)=3(−x)2−4(−x)+2

STEP 9

Simplify the expression for g(−x)g(-x)g(−x).g(−x)=3x2+4x+2 g(-x) = 3x^2 + 4x + 2 g(−x)=3x2+4x+2

STEP 10

Evaluate g(1−t)g(1-t)g(1−t) by substituting 1−t1-t1−t into the function g(x)g(x)g(x).g(1−t)=3(1−t)2−4(1−t)+2 g(1-t) = 3(1-t)^2 - 4(1-t) + 2 g(1−t)=3(1−t)2−4(1−t)+2

STEP 11

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QuestionGiven that $\mathrm{g}(\mathrm{x})=3 \mathrm{x}^{2}-4 \mathrm{x}+2$ , find each of the following. a) $g(0)$ b) $g(-2)$ c) $g(3)$ d) $g(-x)$ e) $g(1-t)$

Evaluate $g(0)$ .
$g(0) = 3(0)^2 - 4(0) + 2 = 2$

Evaluate $g(-2)$ .
$g(-2) = 3(-2)^2 - 4(-2) + 2$

Simplify the expression for $g(-2)$ .
$g(-2) = 3(4) + 8 + 2 = 12 + 8 + 2 = 22$

Evaluate $g(3)$ .
$g(3) = 3(3)^2 - 4(3) + 2$

Simplify the expression for $g(3)$ .
$g(3) = 3(9) - 12 + 2 = 27 - 12 + 2 = 17$

Evaluate $g(-x)$ by substituting $-x$ into the function $g(x)$ .
$g(-x) = 3(-x)^2 - 4(-x) + 2$

Simplify the expression for $g(-x)$ .
$g(-x) = 3x^2 + 4x + 2$

Evaluate $g(1-t)$ by substituting $1-t$ into the function $g(x)$ .
$g(1-t) = 3(1-t)^2 - 4(1-t) + 2$