Function Zeros

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intercepts

Let $f$ be a function with its natural domain.

The domain number $b$ is a zero of $f$, if $f(b) = 0$.

The point on the graph corresponding to the zero $b$ is $(b, f(b)) = (b, 0)$, an intercept.

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

The graph has one intercept, which means that $G$ has one zero, which appears to be $2.4$.

\[ G(2.4) = 0 \]

The graph has another intercept: $(0, 1.7)$. This tells us that $G(0)=1.7$. However, this is not really useful information in function analysis.

If this function was a model for some timed event, then perhaps $t = 0$, would be important to the situation, as initial information. That would be left for the interpretation of the model.

Absent some physical interpretation, when analyzing a function, we are only interested in its zeros, which correspond to intercepts on the horizontal axis.

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 Functions

Some core functions have zeros and others do not.

Constant Functions

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

Power Functions

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.

Root Functions

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

Exponential Functions

Core exponential functions do not have zeros.

Logarithmic Functions

Core logarithmic functions have exactly one zero. The zero is the domain number that makes the inside of the logarithm equal to $1$.

Sine and Cosine

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 $\{ \frac {\pi }{2} + k\pi \, | \, k \in \mathbb {Z} \}$

Absolute Value

The absolute value function has one zero. It is $0$.

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2026-05-23 20:32:58