Questions: Suppose that the functions (s) and (t) are defined for all real numbers (x) as follows [ s(x)=2 x-1 t(x)=6 x ] Write the expressions for ((t cdot s)(x)) and ((t+s)(x)) and evaluate ((t-s)(-2)). [ (t cdot s)(x) =square (t+s)(x) =square (t-s)(-2) =square ]

Suppose that the functions (s) and (t) are defined for all real numbers (x) as follows
[
s(x)=2 x-1
t(x)=6 x
]

Write the expressions for ((t cdot s)(x)) and ((t+s)(x)) and evaluate ((t-s)(-2)).
[
(t cdot s)(x) =square
(t+s)(x) =square
(t-s)(-2) =square
]

Transcript text: Suppose that the functions $s$ and $t$ are defined for all real numbers $x$ as follows \[ \begin{array}{l} s(x)=2 x-1 \\ t(x)=6 x \end{array} \] Write the expressions for $(t \cdot s)(x)$ and $(t+s)(x)$ and evaluate $(t-s)(-2)$. \[ \begin{aligned} (t \cdot s)(x) & =\square \\ (t+s)(x) & =\square \\ (t-s)(-2) & =\square \end{aligned} \]

Solution

Solution Steps

To solve the given problem, we need to perform operations on the functions $ s(x) = 2x - 1 $ and $ t(x) = 6x $.

For $(t \cdot s)(x)$, we multiply the two functions: $ t(x) \times s(x) $.
For $(t+s)(x)$, we add the two functions: $ t(x) + s(x) $.
To evaluate $(t-s)(-2)$, we subtract the functions and then substitute $ x = -2 $.

Step 1: Find the Expression for $(t \cdot s)(x)$

To find the product of the functions $t(x)$ and $s(x)$, we multiply the expressions for each function:

\[ t(x) = 6x, \quad s(x) = 2x - 1 \]

The product $(t \cdot s)(x)$ is given by:

\[ (t \cdot s)(x) = t(x) \cdot s(x) = (6x) \cdot (2x - 1) \]

Distribute $6x$ across the terms in $s(x)$:

\[ (t \cdot s)(x) = 6x \cdot 2x - 6x \cdot 1 = 12x^2 - 6x \]

Step 2: Find the Expression for $(t+s)(x)$

To find the sum of the functions $t(x)$ and $s(x)$, we add the expressions for each function:

\[ t(x) = 6x, \quad s(x) = 2x - 1 \]

The sum $(t+s)(x)$ is given by:

\[ (t+s)(x) = t(x) + s(x) = 6x + (2x - 1) \]