Here is the complete graph of the function \(G(x)\).

• Intercepts of graph: \((3,0)\) and \((7,0)\).
• Zeros of \(g\): \(3\) and \(7\).

Note: The intercepts are points on the graph. They are not zeros of the function. Function zeros are domain numbers. The intercepts on the graph visually encode a function zero as the first coordinate of the intercept.

Note: The \(y\)-intercept has nothing to do with function zeros.

Let \(P(w)\) be a function. The graph of \(y = P(w)\) is displayed below.

The graph appears to have two intercepts, not three. This means the function \(P\) has two zeros.

• Intercepts of graph: \((-5,0)\) and \((5,0)\).
• Zeros of \(P\): \(-5\) and \(5\).

Note: The intercepts are points on the graph. They are not zeros of the function. Function zeros are domain numbers. The intercepts on the graph visually encode a function zero as the first coordinate of the intercept.

Note: \(-8\) is in the domain. However, there is an open/hollow dot at \((-8,0)\). which means it is not there. The dot for \(-8\) is \((-8,4)\). \(-8\) is not a zero of \(P\).

Note: The \(y\)-intercept has nothing to do with function zeros.

Let \(K(z)\) be a function. The graph of \(y = K(z)\) is displayed below.

The graph appears to have no intercepts. That is because the whole graph is not drawn. The arrow tells us that the graph continues to go down to the right.

That means there is an intercept around \((-3.8,0)\).

\(-3.8\) is the only zero of \(K(z)\).

Let \(S(\theta )\) be a function. The graph of \(y = S(\theta )\) is displayed below.

The graph appears to have an infinite number of intercepts.

They occur at even integer.

The zeros of \(S(\theta )\) are

Here is a complete graph of \(K(v)\).

In this graph, \(y=2\) is a horizontal asymptote, which means the graph will slowly approach the asymptote and stay near it.

This tells that eventually the values of \(K\) approach \(2\) and stay near \(2\).

This tells that eventually the values of \(K\) will be away from \(0\).

The graph seems to suggest that after \(15\), the values of \(K\) are no longer near \(0\).

That would give \(K\) four zeros: \(3.9\), \(5.8\), \(10.2\), \(11.6\).

Here is a complete graph of \(p(t)\).

The graph has no horizontal intercepts. So, \(p\) has no zeros.

Core constant functions do not have zeros, unless it is the Zero function. Then every domain number is a zero.

If the power is positive, then core power functions only have one zero and that is \(0\).

If the power is negative, then core power functions do not have zeros.

Core root (radical) functions have only \(0\) as a zero.

Core exponential functions do not have zeros.

Core logarithmic functions have exactly one zero, \(1\).

Sine and Cosine have an infinite number of zeros.

The zeros of \(\sin (\theta )\) are \(\{ k\pi \, | \, k \in \mathbb {Z} \}\)

The zeros of \(\cos (\theta )\) are \(\{ \frac {\pi }{2} + k\pi \, | \, k \in \mathbb {Z} \}\)

The absolute value function has one zero. It is \(0\).