Math / Algebra

QuestionSuppose that the functions $s$ and $t$ are defined for all real numbers $x$ as follows. $\begin{array}{l} s(x)=4 x \\ t(x)=3 x-1 \end{array}$
Write the expressions for $(s \cdot t)(x)$ and $(s+t)(x)$ and evaluate $(s-t)(-2)$ . $\begin{array}{r} (s \cdot t)(x)= \\ (s+t)(x)= \\ (s-t)(-2)= \end{array}$ $\square$ $\square$ $\square$

Studdy Solution

STEP 1

1. We are given the functions $s(x) = 4x$ and $t(x) = 3x - 1$ .
2. We need to find expressions for $(s \cdot t)(x)$ and $(s + t)(x)$ .
3. We need to evaluate $(s - t)(-2)$ .

STEP 2

1. Find the expression for $(s \cdot t)(x)$ .
2. Find the expression for $(s + t)(x)$ .
3. Evaluate $(s - t)(-2)$ .

STEP 3

To find $(s \cdot t)(x)$ , multiply the functions $s(x)$ and $t(x)$ :
$(s \cdot t)(x) = s(x) \cdot t(x) = (4x) \cdot (3x - 1)$

STEP 4

Distribute to simplify the expression:
$(s \cdot t)(x) = 4x \cdot 3x - 4x \cdot 1$ $(s \cdot t)(x) = 12x^2 - 4x$

STEP 5

To find $(s + t)(x)$ , add the functions $s(x)$ and $t(x)$ :
$(s + t)(x) = s(x) + t(x) = 4x + (3x - 1)$

STEP 6

Simplify the expression:
$(s + t)(x) = 4x + 3x - 1$ $(s + t)(x) = 7x - 1$

STEP 7

To evaluate $(s - t)(-2)$ , first find the expression for $(s - t)(x)$ :
$(s - t)(x) = s(x) - t(x) = 4x - (3x - 1)$

STEP 8

Simplify the expression:
$(s - t)(x) = 4x - 3x + 1$ $(s - t)(x) = x + 1$

STEP 9

Substitute $x = -2$ into $(s - t)(x)$ :
$(s - t)(-2) = (-2) + 1$ $(s - t)(-2) = -1$
The expressions and evaluation are:
$\begin{array}{r} (s \cdot t)(x) = 12x^2 - 4x \\ (s + t)(x) = 7x - 1 \\ (s - t)(-2) = -1 \end{array}$

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Studdy Solution

STEP 1

1. We are given the functions $s(x) = 4x$ and $t(x) = 3x - 1$ .
2. We need to find expressions for $(s \cdot t)(x)$ and $(s + t)(x)$ .
3. We need to evaluate $(s - t)(-2)$ .

STEP 2

1. Find the expression for $(s \cdot t)(x)$ .
2. Find the expression for $(s + t)(x)$ .
3. Evaluate $(s - t)(-2)$ .

STEP 3

To find $(s \cdot t)(x)$ , multiply the functions $s(x)$ and $t(x)$ :
$(s \cdot t)(x) = s(x) \cdot t(x) = (4x) \cdot (3x - 1)$

STEP 4

Distribute to simplify the expression:
$(s \cdot t)(x) = 4x \cdot 3x - 4x \cdot 1$ $(s \cdot t)(x) = 12x^2 - 4x$

STEP 5

To find $(s + t)(x)$ , add the functions $s(x)$ and $t(x)$ :
$(s + t)(x) = s(x) + t(x) = 4x + (3x - 1)$

STEP 6

Simplify the expression:
$(s + t)(x) = 4x + 3x - 1$ $(s + t)(x) = 7x - 1$

STEP 7

To evaluate $(s - t)(-2)$ , first find the expression for $(s - t)(x)$ :
$(s - t)(x) = s(x) - t(x) = 4x - (3x - 1)$

STEP 8

Simplify the expression:
$(s - t)(x) = 4x - 3x + 1$ $(s - t)(x) = x + 1$

STEP 9

Substitute $x = -2$ into $(s - t)(x)$ :
$(s - t)(-2) = (-2) + 1$ $(s - t)(-2) = -1$
The expressions and evaluation are:
$\begin{array}{r} (s \cdot t)(x) = 12x^2 - 4x \\ (s + t)(x) = 7x - 1 \\ (s - t)(-2) = -1 \end{array}$

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Studdy Solution

STEP 1

1. We are given the functions s(x)=4x s(x) = 4x s(x)=4x and t(x)=3x−1 t(x) = 3x - 1 t(x)=3x−1.2. We need to find expressions for (s⋅t)(x) (s \cdot t)(x) (s⋅t)(x) and (s+t)(x) (s + t)(x) (s+t)(x).3. We need to evaluate (s−t)(−2) (s - t)(-2) (s−t)(−2).

STEP 2

1. Find the expression for (s⋅t)(x) (s \cdot t)(x) (s⋅t)(x).2. Find the expression for (s+t)(x) (s + t)(x) (s+t)(x).3. Evaluate (s−t)(−2) (s - t)(-2) (s−t)(−2).

STEP 3

To find (s⋅t)(x) (s \cdot t)(x) (s⋅t)(x), multiply the functions s(x) s(x) s(x) and t(x) t(x) t(x):(s⋅t)(x)=s(x)⋅t(x)=(4x)⋅(3x−1) (s \cdot t)(x) = s(x) \cdot t(x) = (4x) \cdot (3x - 1) (s⋅t)(x)=s(x)⋅t(x)=(4x)⋅(3x−1)

STEP 4

Distribute to simplify the expression:(s⋅t)(x)=4x⋅3x−4x⋅1 (s \cdot t)(x) = 4x \cdot 3x - 4x \cdot 1 (s⋅t)(x)=4x⋅3x−4x⋅1 (s⋅t)(x)=12x2−4x (s \cdot t)(x) = 12x^2 - 4x (s⋅t)(x)=12x2−4x

STEP 5

To find (s+t)(x) (s + t)(x) (s+t)(x), add the functions s(x) s(x) s(x) and t(x) t(x) t(x):(s+t)(x)=s(x)+t(x)=4x+(3x−1) (s + t)(x) = s(x) + t(x) = 4x + (3x - 1) (s+t)(x)=s(x)+t(x)=4x+(3x−1)

STEP 6

Simplify the expression:(s+t)(x)=4x+3x−1 (s + t)(x) = 4x + 3x - 1 (s+t)(x)=4x+3x−1 (s+t)(x)=7x−1 (s + t)(x) = 7x - 1 (s+t)(x)=7x−1

STEP 7

To evaluate (s−t)(−2) (s - t)(-2) (s−t)(−2), first find the expression for (s−t)(x) (s - t)(x) (s−t)(x):(s−t)(x)=s(x)−t(x)=4x−(3x−1) (s - t)(x) = s(x) - t(x) = 4x - (3x - 1) (s−t)(x)=s(x)−t(x)=4x−(3x−1)

STEP 8

Simplify the expression:(s−t)(x)=4x−3x+1 (s - t)(x) = 4x - 3x + 1 (s−t)(x)=4x−3x+1 (s−t)(x)=x+1 (s - t)(x) = x + 1 (s−t)(x)=x+1

STEP 9

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1. We are given the functions $s(x) = 4x$ and $t(x) = 3x - 1$ .
2. We need to find expressions for $(s \cdot t)(x)$ and $(s + t)(x)$ .
3. We need to evaluate $(s - t)(-2)$ .

1. Find the expression for $(s \cdot t)(x)$ .
2. Find the expression for $(s + t)(x)$ .
3. Evaluate $(s - t)(-2)$ .

To find $(s \cdot t)(x)$ , multiply the functions $s(x)$ and $t(x)$ :
$(s \cdot t)(x) = s(x) \cdot t(x) = (4x) \cdot (3x - 1)$

Distribute to simplify the expression:
$(s \cdot t)(x) = 4x \cdot 3x - 4x \cdot 1$ $(s \cdot t)(x) = 12x^2 - 4x$

To find $(s + t)(x)$ , add the functions $s(x)$ and $t(x)$ :
$(s + t)(x) = s(x) + t(x) = 4x + (3x - 1)$

Simplify the expression:
$(s + t)(x) = 4x + 3x - 1$ $(s + t)(x) = 7x - 1$

To evaluate $(s - t)(-2)$ , first find the expression for $(s - t)(x)$ :
$(s - t)(x) = s(x) - t(x) = 4x - (3x - 1)$

Simplify the expression:
$(s - t)(x) = 4x - 3x + 1$ $(s - t)(x) = x + 1$