Key Concepts and Formulas

• Composite Function Definition: The composite function $(g \circ f)(x)$ is defined as $g(f(x))$. This means the output of function $f$ becomes the input for function $g$.
• Function Evaluation: To evaluate a function $h(x)$ at a specific value $a$, we substitute $a$ for every occurrence of $x$ in the definition of $h(x)$, i.e., $h(a)$.
• Solving Quadratic Equations: A quadratic equation of the form $ay^2 + by + c = 0$ can be solved using factoring, completing the square, or the quadratic formula.

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Step-by-Step Solution

Step 1: Understand the Given Information and the Goal We are given two functions:

1. $g(x) = x^2 + x - 1$
2. $(g \circ f)(x) = 4x^2 - 10x + 5$

Our objective is to find the value of $f\left(\frac{5}{4}\right)$.

Step 2: Evaluate the Composite Function at the Specific Point We are given the expression for the composite function $(g \circ f)(x)$. To find the value of this composite function when $x = \frac{5}{4}$, we substitute $\frac{5}{4}$ into its expression: $(g \circ f)\left(\frac{5}{4}\right) = 4\left(\frac{5}{4}\right)^2 - 10\left(\frac{5}{4}\right) + 5$ Let's perform the calculations: $(g \circ f)\left(\frac{5}{4}\right) = 4\left(\frac{25}{16}\right) - \frac{50}{4} + 5$ $(g \circ f)\left(\frac{5}{4}\right) = \frac{25}{4} - \frac{50}{4} + 5$ To combine these terms, we use a common denominator of 4: $(g \circ f)\left(\frac{5}{4}\right) = \frac{25}{4} - \frac{50}{4} + \frac{20}{4}$ $(g \circ f)\left(\frac{5}{4}\right) = \frac{25 - 50 + 20}{4}$ $(g \circ f)\left(\frac{5}{4}\right) = \frac{-5}{4}$ From the definition of composite functions, we know that $(g \circ f)\left(\frac{5}{4}\right) = g\left(f\left(\frac{5}{4}\right)\right)$. Therefore, we have: $g\left(f\left(\frac{5}{4}\right)\right) = -\frac{5}{4} \quad (*)$

Step 3: Express the Composite Function in Terms of $f(x)$ We are given $g(x) = x^2 + x - 1$. To find $g(f(x))$, we substitute $f(x)$ in place of $x$ in the expression for $g(x)$: $g(f(x)) = (f(x))^2 + f(x) - 1$

Step 4: Evaluate the Expression from Step 3 at the Specific Point Now, we substitute $x = \frac{5}{4}$ into the expression for $g(f(x))$ derived in Step 3: $g\left(f\left(\frac{5}{4}\right)\right) = \left(f\left(\frac{5}{4}\right)\right)^2 + f\left(\frac{5}{4}\right) - 1$ To simplify this equation, let $y = f\left(\frac{5}{4}\right)$. Then the equation becomes: $g\left(f\left(\frac{5}{4}\right)\right) = y^2 + y - 1 \quad (**)$

Step 5: Equate the Expressions and Solve for $f\left(\frac{5}{4}\right)$ We have two expressions for $g\left(f\left(\frac{5}{4}\right)\right)$: from equation $(*)$ and from equation $(**)$. Equating them, we get: $y^2 + y - 1 = -\frac{5}{4}$ To solve for $y$, we rearrange the equation into a standard quadratic form $ay^2 + by + c = 0$: $y^2 + y - 1 + \frac{5}{4} = 0$ Combine the constant terms: $y^2 + y + \left(-\frac{4}{4} + \frac{5}{4}\right) = 0$ $y^2 + y + \frac{1}{4} = 0$ This is a perfect square trinomial, which can be factored as: $\left(y + \frac{1}{2}\right)^2 = 0$ Taking the square root of both sides: $y + \frac{1}{2} = 0$ Solving for $y$: $y = -\frac{1}{2}$ Since we defined $y = f\left(\frac{5}{4}\right)$, we have: $f\left(\frac{5}{4}\right) = -\frac{1}{2}$

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Common Mistakes & Tips

• Algebraic Errors with Fractions: Be extremely careful when performing arithmetic operations with fractions, especially when squaring or finding common denominators.
• Misinterpreting Composite Functions: Ensure you understand that $(g \circ f)(x) = g(f(x))$ and not $f(g(x))$ or $g(x) \cdot f(x)$.
• Solving Quadratic Equations: If the quadratic equation doesn't appear to be a perfect square, use the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots.

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Summary

The problem involved evaluating a composite function at a specific point and then using the definition of the composite function to set up an equation involving the unknown function $f(x)$. By evaluating both the given composite function and the expression $g(f(x))$ at $x = \frac{5}{4}$, we obtained a quadratic equation in terms of $f\left(\frac{5}{4}\right)$. Solving this quadratic equation yielded the value of $f\left(\frac{5}{4}\right)$.

The final answer is $\boxed{-\frac{1}{2}}$.