To find the function $f \circ g(x)$, we compose $f$ and $g$, which means substituting $g(x)$ into $f(x)$:

Given:

• $f(x) = e^x$
• $g(x) = \ln(x)$

1. Composition of $f$ and $g$:

$f \circ g(x) = f(g(x)) = f(\ln(x)) = e^{\ln(x)}$

Since $e^{\ln(x)} = x$ (by the property of exponents and logarithms), we get: $f \circ g(x) = x$

2. Graph of $f \circ g(x) = x$

The function $f \circ g(x) = x$ is simply the line $y = x$. This is a straight line passing through the origin, with a slope of 1.

3. Domain and Range

• Domain of $f \circ g(x)$: Since $g(x) = \ln(x)$ is defined only for $x > 0$, the domain of $f \circ g(x)$ is also $x > 0$.
• Range of $f \circ g(x)$: Since $f \circ g(x) = x$, the range is all real numbers, $(-\infty, \infty)$.

Summary

• Function: $f \circ g(x) = x$
• Graph: A line $y = x$ (limited to $x > 0$)
• Domain: $x > 0$
• Range: $(-\infty, \infty)$

Would you like to see the graph or need further details?

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1. How does the domain of $g(x)$ affect the composition $f \circ g(x)$?
2. Why is the range of $f \circ g(x)$ different from that of $g(x)$?
3. What would happen to the function if we composed $g \circ f(x)$ instead?
4. What transformations occur when composing functions with exponential and logarithmic forms?
5. How would the composition differ if $g(x)$ were another function, like $x^2$?

Tip: When composing functions, always check if the output of the inner function fits within the input domain of the outer function to define the overall domain correctly.