Find the Domain and Range f(x)=- log of 5-x+9

$f (x) = - \log (5 - x) + 9$

Step 1

Set the argument in $\log (5 - x)$ greater than $0$ to find where the expression is defined.

$5 - x > 0$

Step 2

Solve for $x$ .

Subtract $5$ from both sides of the inequality.

$- x > - 5$

Divide each term in $- x > - 5$ by $- 1$ and simplify.

Divide each term in $- x > - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x < \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x < 5$

Step 3

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

Interval Notation:

$(- \infty, 5)$

Set-Builder Notation:

${x | x < 5}$

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: $(- \infty, 5), {x | x < 5}$

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

Step 6

Find the Domain and Range f(x)=- log of 5-x+9

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