Precalculus Examples

$f (x) = \frac{1}{2} \log (4 x)$

Step 1

Set the argument in $\log (4 x)$ 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.

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Step 2.1

Divide each term in $4 x > 0$ by $4$ .

$\frac{4 x}{4} > \frac{0}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} > \frac{0}{4}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $4$ .

$x > 0$

Step 3

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

Interval Notation:

$(0, \infty)$

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: $(0, \infty), {x | x > 0}$

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

Step 6