Find the Domain and Range f(x) = natural log of -4x+3

$f (x) = \ln (- 4 x) + 3$

Step 1

Set the argument in $\ln (- 4 x)$ greater than $0$ to find where the expression is defined.

$- 4 x > 0$

Step 2

Divide each term in $- 4 x > 0$ by $- 4$ and simplify.

Divide each term in $- 4 x > 0$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{0}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{0}{- 4}$

Divide $x$ by $1$ .

$x < \frac{0}{- 4}$

Simplify the right side.

Divide $0$ by $- 4$ .

$x < 0$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0)$

Set-Builder Notation:

${x | x < 0}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 0), {x | x < 0}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6

Find the Domain and Range f(x) = natural log of -4x+3

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