Math / Algebra

Question $1 \& \quad$ For the function $f(x)=\frac{4 x-12}{x+5}$ , complete the following parts. (a) Find $f(x)$ for $x=-1$ and $d$ , if possible. (b) Find the domain of $f$ . (a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. $\square$ A. $f(-1)=-4$ (Simplify your answer.) B. The value of $f(-1)$ is undefined.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $f(d)=\frac{4 d-12}{d+6}$ (Simplify your answer.) B. The value of $f(d)$ is undefined.

Studdy Solution

STEP 1

1. The function $f(x) = \frac{4x - 12}{x + 5}$ is a rational function.
2. To evaluate $f(x)$ for specific values, substitute those values into the function.
3. The domain of a rational function is all real numbers except where the denominator is zero.

STEP 2

1. Evaluate $f(x)$ for $x = -1$ .
2. Evaluate $f(x)$ for $x = d$ .
3. Determine the domain of the function $f(x)$ .

STEP 3

Substitute $x = -1$ into the function $f(x)$ .
$f(-1) = \frac{4(-1) - 12}{-1 + 5}$

STEP 4

Simplify the numerator and denominator.
$f(-1) = \frac{-4 - 12}{4}$

STEP 5

Combine the terms in the numerator.
$f(-1) = \frac{-16}{4}$

STEP 6

Simplify the fraction.
$f(-1) = -4$

STEP 7

Substitute $x = d$ into the function $f(x)$ .
$f(d) = \frac{4d - 12}{d + 5}$

STEP 8

Identify the values of $x$ that make the denominator zero. Set the denominator equal to zero and solve for $x$ .
$x + 5 = 0$

STEP 9

Solve the equation for $x$ .
$x = -5$

STEP 10

State the domain of $f(x)$ . The domain is all real numbers except $x = -5$ .
$\text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq -5 \}$
Solution: (a) The correct choice is A. $f(-1) = -4$ . (b) The correct choice is A. $f(d) = \frac{4d - 12}{d + 5}$ . The domain of $f(x)$ is all real numbers except $x = -5$ .

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Studdy Solution

STEP 1

1. The function $f(x) = \frac{4x - 12}{x + 5}$ is a rational function.
2. To evaluate $f(x)$ for specific values, substitute those values into the function.
3. The domain of a rational function is all real numbers except where the denominator is zero.

STEP 2

1. Evaluate $f(x)$ for $x = -1$ .
2. Evaluate $f(x)$ for $x = d$ .
3. Determine the domain of the function $f(x)$ .

STEP 3

Substitute $x = -1$ into the function $f(x)$ .
$f(-1) = \frac{4(-1) - 12}{-1 + 5}$

STEP 4

Simplify the numerator and denominator.
$f(-1) = \frac{-4 - 12}{4}$

STEP 5

Combine the terms in the numerator.
$f(-1) = \frac{-16}{4}$

STEP 6

Simplify the fraction.
$f(-1) = -4$

STEP 7

Substitute $x = d$ into the function $f(x)$ .
$f(d) = \frac{4d - 12}{d + 5}$

STEP 8

Identify the values of $x$ that make the denominator zero. Set the denominator equal to zero and solve for $x$ .
$x + 5 = 0$

STEP 9

Solve the equation for $x$ .
$x = -5$

STEP 10

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Studdy Solution

STEP 1

1. The function f(x)=4x−12x+5 f(x) = \frac{4x - 12}{x + 5} f(x)=x+54x−12​ is a rational function.2. To evaluate f(x) f(x) f(x) for specific values, substitute those values into the function.3. The domain of a rational function is all real numbers except where the denominator is zero.

STEP 2

1. Evaluate f(x) f(x) f(x) for x=−1 x = -1 x=−1.2. Evaluate f(x) f(x) f(x) for x=d x = d x=d.3. Determine the domain of the function f(x) f(x) f(x).

STEP 3

Substitute x=−1 x = -1 x=−1 into the function f(x) f(x) f(x).f(−1)=4(−1)−12−1+5 f(-1) = \frac{4(-1) - 12}{-1 + 5} f(−1)=−1+54(−1)−12​

STEP 4

Simplify the numerator and denominator.f(−1)=−4−124 f(-1) = \frac{-4 - 12}{4} f(−1)=4−4−12​

STEP 5

Combine the terms in the numerator.f(−1)=−164 f(-1) = \frac{-16}{4} f(−1)=4−16​

STEP 6

Simplify the fraction.f(−1)=−4 f(-1) = -4 f(−1)=−4

STEP 7

Substitute x=d x = d x=d into the function f(x) f(x) f(x).f(d)=4d−12d+5 f(d) = \frac{4d - 12}{d + 5} f(d)=d+54d−12​

STEP 8

Identify the values of x x x that make the denominator zero. Set the denominator equal to zero and solve for x x x.x+5=0 x + 5 = 0 x+5=0

STEP 9

Solve the equation for x x x.x=−5 x = -5 x=−5

STEP 10

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1. The function $f(x) = \frac{4x - 12}{x + 5}$ is a rational function.
2. To evaluate $f(x)$ for specific values, substitute those values into the function.
3. The domain of a rational function is all real numbers except where the denominator is zero.

1. Evaluate $f(x)$ for $x = -1$ .
2. Evaluate $f(x)$ for $x = d$ .
3. Determine the domain of the function $f(x)$ .

Substitute $x = -1$ into the function $f(x)$ .
$f(-1) = \frac{4(-1) - 12}{-1 + 5}$

Simplify the numerator and denominator.
$f(-1) = \frac{-4 - 12}{4}$

Combine the terms in the numerator.
$f(-1) = \frac{-16}{4}$

Simplify the fraction.
$f(-1) = -4$

Substitute $x = d$ into the function $f(x)$ .
$f(d) = \frac{4d - 12}{d + 5}$

Identify the values of $x$ that make the denominator zero. Set the denominator equal to zero and solve for $x$ .
$x + 5 = 0$

Solve the equation for $x$ .
$x = -5$