To solve the problem, we need to evaluate $(f \circ g) \left( -\frac{1}{5} \right)$, where $f(x) = x^2 - 15$ and $g(x) = 5x + 1$.

Solution Steps

1. Find $g\left(-\frac{1}{5}\right)$: $g\left(-\frac{1}{5}\right) = 5 \cdot \left(-\frac{1}{5}\right) + 1 = -1 + 1 = 0$

2. Substitute into $f(x)$ to find $f(g(x))$: Since $g\left(-\frac{1}{5}\right) = 0$, we need to find $f(0)$. $f(0) = 0^2 - 15 = -15$

3. Conclusion: $(f \circ g) \left( -\frac{1}{5} \right) = -15$

Final Answer

$(f \circ g) \left( -\frac{1}{5} \right) = -15$

Would you like more details on any step, or have additional questions?

Related Questions

1. How do you find the composition of two functions generally?
2. What conditions must be met for the composition $(f \circ g)(x)$ to be defined?
3. What would $(g \circ f)(x)$ be for this problem?
4. How would we solve this if $g(x)$ was a quadratic function?
5. How can we interpret function composition in real-world scenarios?

Tip

In function composition, always evaluate the inner function first and use its result as the input for the outer function.