Finite Math Examples

$f (x) = \log_{2} (3 x - 5)$

Step 1

Set the argument in $\log_{2} (3 x - 5)$ greater than $0$ to find where the expression is defined.

$3 x - 5 > 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Add $5$ to both sides of the inequality.

$3 x > 5$

Step 2.2

Divide each term in $3 x > 5$ by $3$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $3 x > 5$ by $3$ .

$\frac{3 x}{3} > \frac{5}{3}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{5}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{5}{3}$

Step 3

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

Interval Notation:

$(\frac{5}{3}, \infty)$

Set-Builder Notation:

${x | x > \frac{5}{3}}$

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: $(\frac{5}{3}, \infty), {x | x > \frac{5}{3}}$

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

Step 6