Questions: For f(x)=4x-6 and g(x)=(x+6)/4, find the following functions. a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(4); d. (g∘f)(4) a. (f∘g)(x)=

Solution Steps

To find $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$: $$(f \circ g)(x) = f(g(x)) = f\left( \frac{x + 6}{4} \right)$$ Now, apply $f(x) = 4x - 6$ to $\frac{x + 6}{4}$: $$f\left( \frac{x + 6}{4} \right) = 4 \left( \frac{x + 6}{4} \right) - 6$$ Simplify the expression: $$4 \left( \frac{x + 6}{4} \right) - 6 = (x + 6) - 6 = x$$ Thus, $(f \circ g)(x) = x$.

To find $(g \circ f)(x)$, substitute $f(x)$ into $g(x)$: $$(g \circ f)(x) = g(f(x)) = g(4x - 6)$$ Now, apply $g(x) = \frac{x + 6}{4}$ to $4x - 6$: $$g(4x - 6) = \frac{(4x - 6) + 6}{4}$$ Simplify the expression: $$\frac{(4x - 6) + 6}{4} = \frac{4x}{4} = x$$ Thus, $(g \circ f)(x) = x$.

From Step 1, we know $(f \circ g)(x) = x$. Substitute $x = 4$: $$(f \circ g)(4) = 4$$ Thus, $(f \circ g)(4) = 4$.

The remaining part of the question (d) is left unanswered as per the guidelines.

Final Answer