Questions: Suppose that the functions (f) and (g) are defined for all real numbers (x) as follows. (f(x)=x-5) (g(x)=4x-6) Write the expressions for ((f-g)(x)) and ((f cdot g)(x)) and evaluate ((f+g)(1)). ((f-g)(x) =-3x+1) ((f cdot g)(x) =4x^2-26x+30) ((f+g)(1) =-6)

Suppose that the functions (f) and (g) are defined for all real numbers (x) as follows.

(f(x)=x-5)
(g(x)=4x-6)

Write the expressions for ((f-g)(x)) and ((f cdot g)(x)) and evaluate ((f+g)(1)).

((f-g)(x) =-3x+1)
((f cdot g)(x) =4x^2-26x+30)
((f+g)(1) =-6)

Transcript text: Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows. \[ \begin{array}{l} f(x)=x-5 \\ g(x)=4 x-6 \end{array} \] Write the expressions for $(f-g)(x)$ and $(f \cdot g)(x)$ and evaluate $(f+g)(1)$. \[ \begin{aligned} (f-g)(x) & =-3 x+1 \\ (f \cdot g)(x) & =4 x^{2}-26 x+30 \\ (f+g)(1) & =-6 \end{aligned} \]

Solution

Solution Steps

To solve the given problem, we need to perform operations on the functions $ f(x) = x - 5 $ and $ g(x) = 4x - 6 $. First, for $(f-g)(x)$, subtract $ g(x) $ from $ f(x) $. For $(f \cdot g)(x)$, multiply $ f(x) $ by $ g(x) $. Finally, for $(f+g)(1)$, evaluate the sum of $ f(x) $ and $ g(x) $ at $ x = 1 $.

Step 1: Define the Functions

The functions are defined as follows: \[ f(x) = x - 5 \] \[ g(x) = 4x - 6 \]

Step 2: Calculate $(f-g)(x)$

To find $(f-g)(x)$, we subtract $g(x)$ from $f(x)$: \[ (f-g)(x) = f(x) - g(x) = (x - 5) - (4x - 6) = x - 5 - 4x + 6 = -3x + 1 \]

Step 3: Calculate $(f \cdot g)(x)$

To find $(f \cdot g)(x)$, we multiply $f(x)$ by $g(x)$: \[ (f \cdot g)(x) = f(x) \cdot g(x) = (x - 5)(4x - 6) \] Expanding this: \[ = 4x^2 - 6x - 20x + 30 = 4x^2 - 26x + 30 \]

Step 4: Evaluate $(f+g)(1)$

To evaluate $(f+g)(1)$, we first find $f(1)$ and $g(1)$: \[ f(1) = 1 - 5 = -4 \] \[ g(1) = 4(1) - 6 = -2 \] Now, we sum these values: \[ (f+g)(1) = f(1) + g(1) = -4 + (-2) = -6 \]