Horizontal intercept

f(x)=\root(3)(x-11)

f(x)=\root(3)(16x+29)

g(x)=\root(4)(-28x+27)

f(x)=\root(5)(-x)

p(x)=\root(6)(-23-25x)

To find the horizontal intercepts (also known as \(x\)-intercepts) of these functions, we need to set each function equal to zero and solve for \(x\). This is because the \(x\)-intercept occurs where the function \(f(x) = 0\).

1. \( f(x) = \sqrt[3]{x - 11} \)

\[ 0 = \sqrt[3]{x - 11} \]
\[ 0 = x - 11 \]
\[ x = 11 \]

The horizontal intercept is \(x = 11\).

2. \( f(x) = \sqrt[3]{16x + 29} \)

\[ 0 = \sqrt[3]{16x + 29} \]
\[ 0 = 16x + 29 \]
\[ 16x = -29 \]
\[ x = -\frac{29}{16} \]

The horizontal intercept is \(x = -\frac{29}{16}\).

3. \( g(x) = \sqrt[4]{-28x + 27} \)

\[ 0 = \sqrt[4]{-28x + 27} \]
\[ 0 = -28x + 27 \]
\[ -28x = -27 \]
\[ x = \frac{27}{28} \]

The horizontal intercept is \(x = \frac{27}{28}\).

4. \( f(x) = \sqrt[5]{-x} \)

\[ 0 = \sqrt[5]{-x} \]
\[ 0 = -x \]
\[ x = 0 \]

The horizontal intercept is \(x = 0\).

5. \( p(x) = \sqrt[6]{-23 - 25x} \)

\[ 0 = \sqrt[6]{-23 - 25x} \]
\[ 0 = -23 - 25x \]
\[ -25x = 23 \]
\[ x = -\frac{23}{25} \]

The horizontal intercept is \(x = -\frac{23}{25}\).

Do you want more details or have any questions?

Here are 8 related questions to expand on this topic:

1. How do you find vertical intercepts of a function?
2. What is the difference between a horizontal intercept and a vertical intercept?
3. Can a function have more than one horizontal intercept?
4. How do you find the horizontal intercepts of a quadratic function?
5. How do you solve a cubic equation to find its intercepts?
6. What are the implications of horizontal intercepts in real-world problems?
7. How do transformations affect the horizontal intercepts of a function?
8. Can horizontal intercepts be complex numbers?

**Tip:** When solving for intercepts, always make sure to isolate the variable step-by-step to avoid mistakes.

Discover how to find horizontal intercepts (x-intercepts) of root functions like cube roots, fourth roots, fifth roots, and sixth roots. Includes step-by-step solutions and examples for each type of root function.

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